diff --git a/thys/Bicategory/PseudonaturalTransformation.thy b/thys/Bicategory/PseudonaturalTransformation.thy
--- a/thys/Bicategory/PseudonaturalTransformation.thy
+++ b/thys/Bicategory/PseudonaturalTransformation.thy
@@ -1,8201 +1,8201 @@
 (*  Title:       PseudonaturalTransformation
     Author:      Eugene W. Stark <stark@cs.stonybrook.edu>, 2020
     Maintainer:  Eugene W. Stark <stark@cs.stonybrook.edu>
 *)
 
 section "Pseudonatural Transformations"
 
 theory PseudonaturalTransformation
-imports Bicategory.InternalEquivalence Bicategory.InternalAdjunction Bicategory.Pseudofunctor
+imports InternalEquivalence InternalAdjunction Pseudofunctor
 begin
 
   subsection "Definition of Pseudonatural Transformation"
 
   text \<open>
     Pseudonatural transformations are a generalization of natural transformations that is
     appropriate for pseudofunctors.  The ``components'' of a pseudonatural transformation \<open>\<tau>\<close>
     from a pseudofunctor \<open>F\<close> to a pseudofunctor \<open>G\<close> (both from bicategory \<open>C\<close> to \<open>D\<close>),
     are 1-cells \<open>\<guillemotleft>\<tau>\<^sub>0 a : F\<^sub>0 a \<rightarrow>\<^sub>D G\<^sub>0 a\<guillemotright>\<close> associated with the objects of \<open>C\<close>.  Instead of
     ``naturality squares'' that commute on-the-nose, a pseudonatural transformation associates
     an invertible 2-cell \<open>\<guillemotleft>\<tau>\<^sub>1 f : G f \<star>\<^sub>D \<tau>\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0 a' \<star>\<^sub>D F f\<guillemotright>\<close> with each 1-cell \<open>\<guillemotleft>f : a \<rightarrow>\<^sub>C a'\<guillemotright>\<close>
     of \<open>C\<close>.  The 1-cells \<open>\<tau>\<^sub>0 a\<close> and 2-cells \<open>\<tau>\<^sub>1 f\<close> are subject to coherence conditions which
     express that they transform naturally across 2-cells of \<open>C\<close> and behave sensibly with
     respect to objects and horizontal composition.
 
     In formalizing ordinary natural transformations, we found it convenient to treat them
     similarly to functors in that a natural transformation \<open>\<tau> : C \<rightarrow> D\<close> maps arrows of \<open>C\<close>
     to arrows of \<open>D\<close>; the components \<open>\<tau> a\<close> at objects \<open>a\<close> being merely special cases.
     However, it is not possible to take the same approach for pseudofunctors, because it is
     not possible to treat the components \<open>\<tau>\<^sub>0 a\<close> at objects \<open>a\<close> as a special case of the
     components \<open>\<tau>\<^sub>1 f\<close> at 1-cells \<open>f\<close>.  So we have to regard a pseudonatural transformation \<open>\<tau>\<close>
     as consisting of two separate mappings: a mapping \<open>\<tau>\<^sub>0\<close> from objects of \<open>C\<close> to 1-cells
     of \<open>D\<close> and a mapping \<open>\<tau>\<^sub>1\<close> from 1-cells of \<open>C\<close> to invertible 2-cells of \<open>D\<close>.
 
     Pseudonatural transformations are themselves a special case of the more general notion
     of ``lax natural transformations'' between pseudofunctors.  For a lax natural transformation
     \<open>\<tau>\<close>, the 2-cells \<open>\<tau>\<^sub>1 f\<close> are not required to be invertible.  This means that there is a
     distinction between a lax natural transformation with \<open>\<guillemotleft>\<tau>\<^sub>1 f : G f \<star>\<^sub>D \<tau>\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0 a' \<star>\<^sub>D F f\<guillemotright>\<close>
     and an ``op-lax'' natural transformation with \<open>\<guillemotleft>\<tau>\<^sub>1 f : \<tau>\<^sub>0 a' \<star>\<^sub>D F f \<Rightarrow>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a\<guillemotright>\<close>.
     There is some variation in the literature on which direction is considered ``lax'' and
     which is ``op-lax'' and this variation extends as well to the special case of pseudofunctors,
     though in that case it does not result in any essential difference in the notion,
     due to the assumed invertibility.  We have chosen the direction that agrees with
     Leinster \cite{leinster-basic-bicategories}, and with the ``nLab'' article
     \cite{nlab-lax-natural-transformation} on lax natural transformations, but note that the
     ``nLab'' article \cite{nlab-pseudonatural-transformation} on pseudonatural transformations
     seems to make the opposite choice.
   \<close>
 
   locale pseudonatural_transformation =
     C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C +
     D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D +
     F: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F +
     G: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G
   for V\<^sub>C :: "'c comp"                   (infixr "\<cdot>\<^sub>C" 55)
   and H\<^sub>C :: "'c comp"                   (infixr "\<star>\<^sub>C" 53)
   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"       ("\<a>\<^sub>C[_, _, _]")
   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"                   ("\<i>\<^sub>C[_]")
   and src\<^sub>C :: "'c \<Rightarrow> 'c"
   and trg\<^sub>C :: "'c \<Rightarrow> 'c"
   and V\<^sub>D :: "'d comp"                   (infixr "\<cdot>\<^sub>D" 55)
   and H\<^sub>D :: "'d comp"                   (infixr "\<star>\<^sub>D" 53)
   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"       ("\<a>\<^sub>D[_, _, _]")
   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"                   ("\<i>\<^sub>D[_]")
   and src\<^sub>D :: "'d \<Rightarrow> 'd"
   and trg\<^sub>D :: "'d \<Rightarrow> 'd"
   and F :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>F :: "'c * 'c \<Rightarrow> 'd"
   and G :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>G :: "'c * 'c \<Rightarrow> 'd"
   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"
   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" +
   assumes map\<^sub>0_in_hhom: "C.obj a \<Longrightarrow> \<guillemotleft>\<tau>\<^sub>0 a : src\<^sub>D (F a) \<rightarrow>\<^sub>D src\<^sub>D (G a)\<guillemotright>"
   and map\<^sub>1_in_vhom: "C.ide f \<Longrightarrow> \<guillemotleft>\<tau>\<^sub>1 f : G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f) \<Rightarrow>\<^sub>D \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f\<guillemotright>"
   and ide_map\<^sub>0_obj: "C.obj a \<Longrightarrow> D.ide (\<tau>\<^sub>0 a)"
   and iso_map\<^sub>1_ide: "C.ide f \<Longrightarrow> D.iso (\<tau>\<^sub>1 f)"
   and naturality: "C.arr \<mu> \<Longrightarrow>
                      \<tau>\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) = (\<tau>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<tau>\<^sub>1 (C.dom \<mu>)"
   and respects_unit: "C.obj a \<Longrightarrow>
                         (\<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0 a] = \<tau>\<^sub>1 a \<cdot>\<^sub>D (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a)"
   and respects_hcomp:
         "\<lbrakk> C.ide f; C.ide g; src\<^sub>C g = trg\<^sub>C f \<rbrakk> \<Longrightarrow>
              (\<tau>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D (\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
              D.inv \<a>\<^sub>D[G g, \<tau>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 (src\<^sub>C f)]
                = \<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f))"
   begin
 
     lemma map\<^sub>0_in_hom [intro]:
     assumes "C.obj a"
     shows "\<guillemotleft>\<tau>\<^sub>0 a : F.map\<^sub>0 a \<rightarrow>\<^sub>D G.map\<^sub>0 a\<guillemotright>"
     and "\<guillemotleft>\<tau>\<^sub>0 a : \<tau>\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0 a\<guillemotright>"
       using assms map\<^sub>0_in_hhom [of a]
        apply fastforce
       using assms ide_map\<^sub>0_obj [of a] by fast
 
     lemma map\<^sub>0_simps [simp]:
     assumes "C.obj a"
     shows "D.ide (\<tau>\<^sub>0 a)"
     and "src\<^sub>D (\<tau>\<^sub>0 a) = F.map\<^sub>0 a" and "trg\<^sub>D (\<tau>\<^sub>0 a) = G.map\<^sub>0 a"
       using assms map\<^sub>0_in_hom iso_map\<^sub>1_ide ide_map\<^sub>0_obj by auto
 
     lemma map\<^sub>1_in_hom [intro]:
     assumes "C.ide f"
     shows "\<guillemotleft>\<tau>\<^sub>1 f : F.map\<^sub>0 (src\<^sub>C f) \<rightarrow>\<^sub>D G.map\<^sub>0 (trg\<^sub>C f)\<guillemotright>"
     and "\<guillemotleft>\<tau>\<^sub>1 f : G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f) \<Rightarrow>\<^sub>D \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f\<guillemotright>"
       using assms map\<^sub>1_in_vhom [of f] D.vconn_implies_hpar(1-2) by auto
 
     lemma map\<^sub>1_simps [simp]:
     assumes "C.ide f"
     shows "D.arr (\<tau>\<^sub>1 f)"
     and "src\<^sub>D (\<tau>\<^sub>1 f) = F.map\<^sub>0 (src\<^sub>C f)"
     and "trg\<^sub>D (\<tau>\<^sub>1 f) = G.map\<^sub>0 (trg\<^sub>C f)"
     and "D.dom (\<tau>\<^sub>1 f) = G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f)"
     and "D.cod (\<tau>\<^sub>1 f) = \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f"
       using assms map\<^sub>1_in_hom iso_map\<^sub>1_ide by auto
 
     lemma inv_naturality:
     assumes "C.arr \<mu>"
     shows "(G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 (C.dom \<mu>)) =
            D.inv (\<tau>\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D (\<tau>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>)"
       using assms naturality iso_map\<^sub>1_ide D.invert_opposite_sides_of_square by simp
 
   end
 
   subsection "Identity Pseudonatural Transformation"
 
   locale identity_pseudonatural_transformation =
     C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C +
     D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D +
     F: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F
   for V\<^sub>C :: "'c comp"                   (infixr "\<cdot>\<^sub>C" 55)
   and H\<^sub>C :: "'c comp"                   (infixr "\<star>\<^sub>C" 53)
   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"       ("\<a>\<^sub>C[_, _, _]")
   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"                   ("\<i>\<^sub>C[_]")
   and src\<^sub>C :: "'c \<Rightarrow> 'c"
   and trg\<^sub>C :: "'c \<Rightarrow> 'c"
   and V\<^sub>D :: "'d comp"                   (infixr "\<cdot>\<^sub>D" 55)
   and H\<^sub>D :: "'d comp"                   (infixr "\<star>\<^sub>D" 53)
   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"       ("\<a>\<^sub>D[_, _, _]")
   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"                   ("\<i>\<^sub>D[_]")
   and src\<^sub>D :: "'d \<Rightarrow> 'd"
   and trg\<^sub>D :: "'d \<Rightarrow> 'd"
   and F :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>F :: "'c * 'c \<Rightarrow> 'd"
   begin
 
     abbreviation (input) map\<^sub>0
     where "map\<^sub>0 a \<equiv> F.map\<^sub>0 a"
 
     abbreviation (input) map\<^sub>1
     where "map\<^sub>1 f \<equiv> \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]"
 
     sublocale pseudonatural_transformation
                 V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F F \<Phi>\<^sub>F map\<^sub>0 map\<^sub>1
     proof
       show 1: "\<And>a. C.obj a \<Longrightarrow> D.ide (F.map\<^sub>0 a)"
         using F.map\<^sub>0_simps(1) by blast
       show "\<And>f. C.ide f \<Longrightarrow> D.iso (map\<^sub>1 f)"
         by auto
       show "\<And>a. C.obj a \<Longrightarrow> \<guillemotleft>F.map\<^sub>0 a : src\<^sub>D (F a) \<rightarrow>\<^sub>D src\<^sub>D (F a)\<guillemotright>"
         by (metis C.obj_def' D.in_hhomI D.src.preserves_arr F.map\<^sub>0_def F.map\<^sub>0_simps(2)
             F.map\<^sub>0_simps(3) F.preserves_reflects_arr)
       show "\<And>f. C.ide f \<Longrightarrow> \<guillemotleft>map\<^sub>1 f : F f \<star>\<^sub>D F.map\<^sub>0 (src\<^sub>C f) \<Rightarrow>\<^sub>D F.map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f\<guillemotright>"
         by simp
       show "\<And>\<mu>. C.arr \<mu> \<Longrightarrow>
                   (map\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D (F \<mu> \<star>\<^sub>D F.map\<^sub>0 (src\<^sub>C \<mu>))
                     = (F.map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D map\<^sub>1 (C.dom \<mu>)"
         by (metis D.comp_assoc D.lunit'_naturality D.runit_naturality F.preserves_arr
             F.preserves_cod F.preserves_dom F.preserves_src F.preserves_trg)
       show "\<And>f g. \<lbrakk>C.ide f; C.ide g; src\<^sub>C g = trg\<^sub>C f\<rbrakk> \<Longrightarrow>
                     (F.map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                     \<a>\<^sub>D[F.map\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                     (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<cdot>\<^sub>D \<r>\<^sub>D[F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                     D.inv \<a>\<^sub>D[F g, F.map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                     (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[F g, F f, F.map\<^sub>0 (src\<^sub>C f)]
                       = (\<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D \<r>\<^sub>D[F (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D F.map\<^sub>0 (src\<^sub>C f))"
       proof -
         fix f g
         assume f: "C.ide f" and g: "C.ide g" and fg: "src\<^sub>C g = trg\<^sub>C f"
         have "(F.map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
               \<a>\<^sub>D[F.map\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
               (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<cdot>\<^sub>D \<r>\<^sub>D[F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
               D.inv \<a>\<^sub>D[F g, F.map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
               (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]) \<cdot>\<^sub>D
               \<a>\<^sub>D[F g, F f, F.map\<^sub>0 (src\<^sub>C f)]
                 = (F.map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                   (\<a>\<^sub>D[F.map\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                   (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f)) \<cdot>\<^sub>D
                   (\<r>\<^sub>D[F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                   (D.inv \<a>\<^sub>D[F g, F.map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                   (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f])) \<cdot>\<^sub>D
                   (F g \<star>\<^sub>D \<r>\<^sub>D[F f]) \<cdot>\<^sub>D
                   \<a>\<^sub>D[F g, F f, F.map\<^sub>0 (src\<^sub>C f)]"
           using f g fg D.whisker_right [of "F f"] D.whisker_left [of "F g"] D.comp_assoc
           by simp
         also have "... = (F.map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                          \<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<cdot>\<^sub>D
                          ((\<r>\<^sub>D[F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f)) \<cdot>\<^sub>D
                          \<r>\<^sub>D[F g \<star>\<^sub>D F f]"
         proof -
           have "D.inv \<a>\<^sub>D[F g, F.map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f]) = \<r>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f"
           proof -
             have "D.inv \<a>\<^sub>D[F g, F.map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f]) =
                   D.inv ((F g \<star>\<^sub>D \<l>\<^sub>D[F f]) \<cdot>\<^sub>D \<a>\<^sub>D[F g, F.map\<^sub>0 (src\<^sub>C g), F f])"
               using f g fg 1 D.inv_comp by auto
             also have "... = D.inv (\<r>\<^sub>D[F g] \<star>\<^sub>D F f)"
               using f g fg D.triangle [of "F f" "F g"] by simp
             also have "... = \<r>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f"
               using f g fg by simp
             finally show ?thesis by blast
           qed
           moreover have "(F g \<star>\<^sub>D \<r>\<^sub>D[F f]) \<cdot>\<^sub>D \<a>\<^sub>D[F g, F f, F.map\<^sub>0 (src\<^sub>C f)] = \<r>\<^sub>D[F g \<star>\<^sub>D F f]"
             using f g fg D.runit_hcomp(1) by simp
           moreover have "\<a>\<^sub>D[F.map\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f)
                            = \<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f]"
             using f g fg D.lunit_hcomp(4) by simp
           ultimately show ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = ((F.map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f]) \<cdot>\<^sub>D \<r>\<^sub>D[F g \<star>\<^sub>D F f]"
         proof -
           have "((\<r>\<^sub>D[F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<r>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f)) \<cdot>\<^sub>D \<r>\<^sub>D[F g \<star>\<^sub>D F f] = \<r>\<^sub>D[F g \<star>\<^sub>D F f]"
           proof -
             have "((\<r>\<^sub>D[F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<r>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f)) \<cdot>\<^sub>D \<r>\<^sub>D[F g \<star>\<^sub>D F f] =
                    (\<r>\<^sub>D[F g] \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<r>\<^sub>D[F g \<star>\<^sub>D F f]"
               using f g fg D.whisker_right by simp
             also have "... = (F g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<r>\<^sub>D[F g \<star>\<^sub>D F f]"
               using f g fg D.comp_arr_inv' by simp
             also have "... = \<r>\<^sub>D[F g \<star>\<^sub>D F f]"
               using f g fg D.comp_cod_arr by simp
             finally show ?thesis by blast
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = \<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D \<Phi>\<^sub>F (g, f) \<cdot>\<^sub>D \<r>\<^sub>D[F g \<star>\<^sub>D F f]"
           using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char F.FF_def
                 D.lunit'_naturality [of "\<Phi>\<^sub>F (g, f)"] D.comp_assoc
           by simp
         also have "... = (\<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D \<r>\<^sub>D[F (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D F.map\<^sub>0 (src\<^sub>C f))"
           using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char F.FF_def
                 D.runit_naturality [of "\<Phi>\<^sub>F (g, f)"] D.comp_assoc
           by simp
         finally show "(F.map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                       \<a>\<^sub>D[F.map\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                       (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<cdot>\<^sub>D \<r>\<^sub>D[F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                       D.inv \<a>\<^sub>D[F g, F.map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                       (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]) \<cdot>\<^sub>D
                       \<a>\<^sub>D[F g, F f, F.map\<^sub>0 (src\<^sub>C f)]
                         = (\<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D \<r>\<^sub>D[F (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D F.map\<^sub>0 (src\<^sub>C f))"
           by simp
       qed
       show "\<And>a. C.obj a \<Longrightarrow>
                   (F.map\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F.map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[F.map\<^sub>0 a] =
                   map\<^sub>1 a \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D F.map\<^sub>0 a)"
       proof -
         fix a
         assume a: "C.obj a"
         have "(F.map\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F.map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[F.map\<^sub>0 a] =
               (F.map\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F.map\<^sub>0 a] \<cdot>\<^sub>D \<r>\<^sub>D[F.map\<^sub>0 a]"
           using a 1 D.unitor_coincidence by simp
         also have "... = ((F.map\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F.map\<^sub>0 a]) \<cdot>\<^sub>D \<r>\<^sub>D[F.map\<^sub>0 a]"
           using D.comp_assoc by simp
         also have "... = \<l>\<^sub>D\<^sup>-\<^sup>1[F a] \<cdot>\<^sub>D F.unit a \<cdot>\<^sub>D \<r>\<^sub>D[F.map\<^sub>0 a]"
           using a 1 D.lunit'_naturality [of "F.unit a"] D.comp_assoc by simp
         also have "... = map\<^sub>1 a \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D F.map\<^sub>0 a)"
           using a 1 D.runit_naturality [of "F.unit a"] D.comp_assoc by simp
         finally show "(F.map\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F.map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[F.map\<^sub>0 a] =
                       map\<^sub>1 a \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D F.map\<^sub>0 a)"
           by blast
       qed
     qed
 
     lemma is_pseudonatural_transformation:
     shows "pseudonatural_transformation
              V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F F \<Phi>\<^sub>F map\<^sub>0 map\<^sub>1"
       ..
 
   end
 
   subsection "Composite Pseudonatural Transformation"
 
   text \<open>
     A pseudonatural transformation \<open>\<sigma>\<close> from \<open>F\<close> to \<open>G\<close> and a pseudonatural transformation \<open>\<rho>\<close>
     from \<open>G\<close> to \<open>H\<close> can be composed to obtain a pseudonatural transformation \<open>\<tau>\<close> from
     \<open>F\<close> to \<open>H\<close>.  The components at objects are composed via horizontal composition.
     Composing the components at 1-cells is straightforward, but is formally complicated by
     the need for associativities.  The additional complexity leads to somewhat lengthy
     proofs of the coherence conditions.  This issue only gets worse as we consider additional
     constructions on pseudonatural transformations.
   \<close>
 
   locale composite_pseudonatural_transformation =
     C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C +
     D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D +
     F: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F +
     G: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G +
     H: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H +
     \<sigma>: pseudonatural_transformation
          V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<sigma>\<^sub>0 \<sigma>\<^sub>1 +
     \<rho>: pseudonatural_transformation
          V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<rho>\<^sub>0 \<rho>\<^sub>1
   for V\<^sub>C :: "'c comp"                   (infixr "\<cdot>\<^sub>C" 55)
   and H\<^sub>C :: "'c comp"                   (infixr "\<star>\<^sub>C" 53)
   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"       ("\<a>\<^sub>C[_, _, _]")
   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"                   ("\<i>\<^sub>C[_]")
   and src\<^sub>C :: "'c \<Rightarrow> 'c"
   and trg\<^sub>C :: "'c \<Rightarrow> 'c"
   and V\<^sub>D :: "'d comp"                   (infixr "\<cdot>\<^sub>D" 55)
   and H\<^sub>D :: "'d comp"                   (infixr "\<star>\<^sub>D" 53)
   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"       ("\<a>\<^sub>D[_, _, _]")
   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"                   ("\<i>\<^sub>D[_]")
   and src\<^sub>D :: "'d \<Rightarrow> 'd"
   and trg\<^sub>D :: "'d \<Rightarrow> 'd"
   and F :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>F :: "'c * 'c \<Rightarrow> 'd"
   and G :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>G :: "'c * 'c \<Rightarrow> 'd"
   and H :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>H :: "'c * 'c \<Rightarrow> 'd"
   and \<sigma>\<^sub>0 :: "'c \<Rightarrow> 'd"
   and \<sigma>\<^sub>1 :: "'c \<Rightarrow> 'd"
   and \<rho>\<^sub>0 :: "'c \<Rightarrow> 'd"
   and \<rho>\<^sub>1 :: "'c \<Rightarrow> 'd"
   begin
 
     definition map\<^sub>0
     where "map\<^sub>0 a = \<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a"
 
     definition map\<^sub>1
     where "map\<^sub>1 f = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C f), \<sigma>\<^sub>0 (trg\<^sub>C f), F f] \<cdot>\<^sub>D
                     (\<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                     (\<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]"
 
     sublocale pseudonatural_transformation
       V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F H \<Phi>\<^sub>H map\<^sub>0 map\<^sub>1
     proof
       show "\<And>a. C.obj a \<Longrightarrow> D.ide (map\<^sub>0 a)"
         unfolding map\<^sub>0_def by fastforce
       show "\<And>a. C.obj a \<Longrightarrow> \<guillemotleft>map\<^sub>0 a : src\<^sub>D (F a) \<rightarrow>\<^sub>D src\<^sub>D (H a)\<guillemotright>"
         unfolding map\<^sub>0_def
         using \<rho>.map\<^sub>0_in_hhom \<sigma>.map\<^sub>0_in_hhom by blast
       show "\<And>f. C.ide f \<Longrightarrow> \<guillemotleft>map\<^sub>1 f : H f \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f) \<Rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f\<guillemotright>"
         by (unfold map\<^sub>1_def map\<^sub>0_def, intro D.comp_in_homI) auto
       show "\<And>f. C.ide f \<Longrightarrow> D.iso (map\<^sub>1 f)"
         unfolding map\<^sub>1_def
         using \<rho>.ide_map\<^sub>0_obj \<rho>.iso_map\<^sub>1_ide \<sigma>.ide_map\<^sub>0_obj \<sigma>.iso_map\<^sub>1_ide
         by (intro D.isos_compose) auto
       show "\<And>\<mu>. C.arr \<mu> \<Longrightarrow> map\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (H \<mu> \<star>\<^sub>D map\<^sub>0 (src\<^sub>C \<mu>))
                                = (map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D map\<^sub>1 (C.dom \<mu>)"
       proof -
         fix \<mu>
         assume \<mu>: "C.arr \<mu>"
         have "map\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (H \<mu> \<star>\<^sub>D map\<^sub>0 (src\<^sub>C \<mu>))
                 = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), \<sigma>\<^sub>0 (trg\<^sub>C \<mu>), F (C.cod \<mu>)] \<cdot>\<^sub>D
                   (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), G (C.cod \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)] \<cdot>\<^sub>D
                   (\<rho>\<^sub>1 (C.cod \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D
                   \<a>\<^sub>D\<^sup>-\<^sup>1[H (C.cod \<mu>), \<rho>\<^sub>0 (src\<^sub>C \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)] \<cdot>\<^sub>D
                   (H \<mu> \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>))"
           unfolding map\<^sub>1_def map\<^sub>0_def
           using \<mu> D.comp_assoc by simp
         also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), \<sigma>\<^sub>0 (trg\<^sub>C \<mu>), F (C.cod \<mu>)] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), G (C.cod \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)] \<cdot>\<^sub>D
                          ((\<rho>\<^sub>1 (C.cod \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D
                          ((H \<mu> \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C \<mu>)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>))) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (C.dom \<mu>), \<rho>\<^sub>0 (src\<^sub>C \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)]"
           using \<mu> D.assoc'_naturality [of "H \<mu>" "\<rho>\<^sub>0 (src\<^sub>C \<mu>)" "\<sigma>\<^sub>0 (src\<^sub>C \<mu>)"] D.comp_assoc by simp
         also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), \<sigma>\<^sub>0 (trg\<^sub>C \<mu>), F (C.cod \<mu>)] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), G (C.cod \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)] \<cdot>\<^sub>D
                          ((\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>))) \<cdot>\<^sub>D
                          (\<rho>\<^sub>1 (C.dom \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (C.dom \<mu>), \<rho>\<^sub>0 (src\<^sub>C \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)]"
         proof -
           have "(\<rho>\<^sub>1 (C.cod \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D ((H \<mu> \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C \<mu>)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) =
                 \<rho>\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (H \<mu> \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C \<mu>)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)"
             using \<mu> D.whisker_right by simp
           also have "... = (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu>) \<cdot>\<^sub>D \<rho>\<^sub>1 (C.dom \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)"
             using \<mu> \<rho>.naturality by simp
           also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D
                            (\<rho>\<^sub>1 (C.dom \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>))"
             using \<mu> D.whisker_right by simp
           finally have
               "(\<rho>\<^sub>1 (C.cod \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D ((H \<mu> \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C \<mu>)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) =
                ((\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D (\<rho>\<^sub>1 (C.dom \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>))"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), \<sigma>\<^sub>0 (trg\<^sub>C \<mu>), F (C.cod \<mu>)] \<cdot>\<^sub>D
                          ((\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>))) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), G (C.dom \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)] \<cdot>\<^sub>D
                          (\<rho>\<^sub>1 (C.dom \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (C.dom \<mu>), \<rho>\<^sub>0 (src\<^sub>C \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)]"
           using \<mu> D.assoc_naturality [of "\<rho>\<^sub>0 (trg\<^sub>C \<mu>)" "G \<mu>" "\<sigma>\<^sub>0 (src\<^sub>C \<mu>)"] D.comp_assoc
           by simp
         also have "... = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), \<sigma>\<^sub>0 (trg\<^sub>C \<mu>), F (C.cod \<mu>)] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>)) \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.dom \<mu>)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), G (C.dom \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)] \<cdot>\<^sub>D
                          (\<rho>\<^sub>1 (C.dom \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (C.dom \<mu>), \<rho>\<^sub>0 (src\<^sub>C \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)]"
         proof -
           have "(\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) =
                 \<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (G \<mu> \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>))"
             using \<mu> D.whisker_left by simp
           also have "... = \<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D (\<sigma>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<sigma>\<^sub>1 (C.dom \<mu>)"
             using \<mu> \<sigma>.naturality by simp
           also have "... = (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.dom \<mu>))"
             using \<mu> D.whisker_left by simp
           finally have "(\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) =
                         (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.dom \<mu>))"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C \<mu>)) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), \<sigma>\<^sub>0 (trg\<^sub>C \<mu>), F (C.dom \<mu>)] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D \<sigma>\<^sub>1 (C.dom \<mu>)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C \<mu>), G (C.dom \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)] \<cdot>\<^sub>D
                          (\<rho>\<^sub>1 (C.dom \<mu>) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (C.dom \<mu>), \<rho>\<^sub>0 (src\<^sub>C \<mu>), \<sigma>\<^sub>0 (src\<^sub>C \<mu>)]"
           using \<mu> D.assoc'_naturality [of "\<rho>\<^sub>0 (trg\<^sub>C \<mu>)" "\<sigma>\<^sub>0 (trg\<^sub>C \<mu>)" "F \<mu>"] D.comp_assoc
           by simp
         also have "... = (map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D map\<^sub>1 (C.dom \<mu>)"
           unfolding map\<^sub>1_def map\<^sub>0_def
           using \<mu> by simp
         finally show "map\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (H \<mu> \<star>\<^sub>D map\<^sub>0 (src\<^sub>C \<mu>)) =
                       (map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D map\<^sub>1 (C.dom \<mu>)"
           by blast
       qed
       show "\<And>a. C.obj a \<Longrightarrow> (map\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a]
                                = map\<^sub>1 a \<cdot>\<^sub>D (H.unit a \<star>\<^sub>D map\<^sub>0 a)"
       proof -
         fix a
         assume a: "C.obj a"
         have "map\<^sub>1 a \<cdot>\<^sub>D (H.unit a \<star>\<^sub>D map\<^sub>0 a)
                 = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, \<sigma>\<^sub>0 a, F a] \<cdot>\<^sub>D
                   (\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>1 a) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<rho>\<^sub>0 a, G a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                   (\<rho>\<^sub>1 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                   \<a>\<^sub>D\<^sup>-\<^sup>1[H a, \<rho>\<^sub>0 a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                   (H.unit a \<star>\<^sub>D \<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a)"
           unfolding map\<^sub>1_def map\<^sub>0_def
           using a C.obj_simps D.comp_assoc by simp
         also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, \<sigma>\<^sub>0 a, F a] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>1 a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 a, G a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                          ((\<rho>\<^sub>1 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                          ((H.unit a \<star>\<^sub>D \<rho>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H.map\<^sub>0 a, \<rho>\<^sub>0 a, \<sigma>\<^sub>0 a]"
           using a C.obj_simps D.assoc'_naturality [of "H.unit a" "\<rho>\<^sub>0 a" "\<sigma>\<^sub>0 a"] D.comp_assoc
           by auto
         also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, \<sigma>\<^sub>0 a, F a] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>1 a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<rho>\<^sub>0 a, G a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                          ((\<rho>\<^sub>0 a \<star>\<^sub>D G.unit a) \<star>\<^sub>D \<sigma>\<^sub>0 a)) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H.map\<^sub>0 a, \<rho>\<^sub>0 a, \<sigma>\<^sub>0 a]"
         proof -
           have "(\<rho>\<^sub>1 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D ((H.unit a \<star>\<^sub>D \<rho>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a)
                    = \<rho>\<^sub>1 a \<cdot>\<^sub>D (H.unit a \<star>\<^sub>D \<rho>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a"
             using a C.obj_simps D.whisker_right
             by (metis C.objE D.hcomp_simps(4) D.hseqI' D.ideD(1) D.ideD(3) D.seqI
                 H.unit_simps(1) H.unit_simps(2) H.unit_simps(5)
                 \<rho>.ide_map\<^sub>0_obj \<rho>.map\<^sub>0_simps(3) \<rho>.map\<^sub>1_simps(1) \<rho>.map\<^sub>1_simps(4) \<sigma>.ide_map\<^sub>0_obj)
           also have "... = (\<rho>\<^sub>0 a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a"
             using a C.obj_simps \<rho>.respects_unit D.comp_assoc by simp
           also have "... = ((\<rho>\<^sub>0 a \<star>\<^sub>D G.unit a) \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a)"
             using a C.obj_simps D.whisker_right
             by (metis C.objE D.hcomp_simps(4) D.hseqI' D.ideD(1) D.ideD(3) D.seqI D.trg_cod
                 H.unit_simps(1-2,5) H.\<i>_simps(1,3,5) \<rho>.ide_map\<^sub>0_obj \<rho>.map\<^sub>0_simps(3)
                 \<rho>.map\<^sub>1_simps(1,4) \<rho>.respects_unit \<sigma>.ide_map\<^sub>0_obj)
           finally have "(\<rho>\<^sub>1 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D ((H.unit a \<star>\<^sub>D \<rho>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a) =
                         ((\<rho>\<^sub>0 a \<star>\<^sub>D G.unit a) \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, \<sigma>\<^sub>0 a, F a] \<cdot>\<^sub>D
                          ((\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>1 a) \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<sigma>\<^sub>0 a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 a, G.map\<^sub>0 a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H.map\<^sub>0 a, \<rho>\<^sub>0 a, \<sigma>\<^sub>0 a]"
           using a D.assoc_naturality [of "\<rho>\<^sub>0 a" "G.unit a" "\<sigma>\<^sub>0 a"] D.comp_assoc by auto
         also have "... = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, \<sigma>\<^sub>0 a, F a] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a \<star>\<^sub>D F.unit a)) \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 a \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<sigma>\<^sub>0 a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 a, G.map\<^sub>0 a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H.map\<^sub>0 a, \<rho>\<^sub>0 a, \<sigma>\<^sub>0 a]"
         proof -
           have "(\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>1 a) \<cdot>\<^sub>D (\<rho>\<^sub>0 a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<sigma>\<^sub>0 a)
                    = \<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>1 a \<cdot>\<^sub>D (G.unit a \<star>\<^sub>D \<sigma>\<^sub>0 a)"
             using a D.whisker_left [of "\<rho>\<^sub>0 a" "\<sigma>\<^sub>1 a" "G.unit a \<star>\<^sub>D \<sigma>\<^sub>0 a"] by force
           also have "... = \<rho>\<^sub>0 a \<star>\<^sub>D (\<sigma>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<sigma>\<^sub>0 a]"
             using a \<sigma>.respects_unit by simp
           also have "... = (\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D (\<rho>\<^sub>0 a \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<sigma>\<^sub>0 a])"
             using a D.whisker_left C.obj_simps by fastforce
           finally have "(\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>1 a) \<cdot>\<^sub>D (\<rho>\<^sub>0 a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<sigma>\<^sub>0 a) =
                         (\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D (\<rho>\<^sub>0 a \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<sigma>\<^sub>0 a])"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = ((\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, \<sigma>\<^sub>0 a, F.map\<^sub>0 a] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 a \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<sigma>\<^sub>0 a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 a, G.map\<^sub>0 a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H.map\<^sub>0 a, \<rho>\<^sub>0 a, \<sigma>\<^sub>0 a]"
           using a D.assoc'_naturality [of "\<rho>\<^sub>0 a" "\<sigma>\<^sub>0 a" "F.unit a"] D.comp_assoc
           by fastforce
         also have "... = ((\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, \<sigma>\<^sub>0 a, F.map\<^sub>0 a] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 a \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 a])) \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 a \<star>\<^sub>D \<l>\<^sub>D[\<sigma>\<^sub>0 a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 a, G.map\<^sub>0 a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H.map\<^sub>0 a, \<rho>\<^sub>0 a, \<sigma>\<^sub>0 a]"
           using a D.whisker_left D.whisker_right D.comp_assoc by simp
         also have "... = ((\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D
                          \<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                          (\<r>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, G.map\<^sub>0 a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 a, G.map\<^sub>0 a, \<sigma>\<^sub>0 a]) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a)) \<cdot>\<^sub>D
                          \<l>\<^sub>D[\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a]"
           using a D.lunit_hcomp(3) [of "\<rho>\<^sub>0 a" "\<sigma>\<^sub>0 a"] D.runit_hcomp(2) [of "\<rho>\<^sub>0 a" "\<sigma>\<^sub>0 a"]
                 D.triangle' [of "\<rho>\<^sub>0 a" "\<sigma>\<^sub>0 a"] D.comp_assoc
           by auto
        also have "... = ((\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a]"
        proof -
          have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a, G.map\<^sub>0 a, \<sigma>\<^sub>0 a] \<cdot>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 a, G.map\<^sub>0 a, \<sigma>\<^sub>0 a]
                  = (\<rho>\<^sub>0 a \<star>\<^sub>D G.map\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a"
            using a D.comp_inv_arr'
            by (metis C.obj_def' D.comp_assoc_assoc'(2) G.map\<^sub>0_def G.map\<^sub>0_simps(1)
                G.preserves_trg G.weak_arrow_of_homs_axioms \<rho>.ide_map\<^sub>0_obj \<rho>.map\<^sub>0_simps(2)
                \<sigma>.ide_map\<^sub>0_obj \<sigma>.map\<^sub>0_simps(3) horizontal_homs.objE weak_arrow_of_homs_def)
          moreover have "((\<rho>\<^sub>0 a \<star>\<^sub>D G.map\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a)
                           = (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a)"
            using a D.comp_cod_arr D.whisker_right
            by (metis D.runit'_simps(1) D.runit'_simps(5) G.map\<^sub>0_def \<rho>.ide_map\<^sub>0_obj
                \<rho>.map\<^sub>0_simps(2) \<sigma>.ide_map\<^sub>0_obj)
          moreover have "(\<r>\<^sub>D[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D (\<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<star>\<^sub>D \<sigma>\<^sub>0 a) = \<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a"
            using a D.whisker_right D.comp_arr_inv' D.R.components_are_iso
            by (metis D.ideD(1) D.iso_runit D.runit_simps(5) \<rho>.ide_map\<^sub>0_obj \<sigma>.ide_map\<^sub>0_obj)
          moreover have "(\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a) \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a] = \<l>\<^sub>D[\<rho>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a]"
            using a D.comp_cod_arr \<open>\<And>a. C.obj a \<Longrightarrow> D.ide (map\<^sub>0 a)\<close> map\<^sub>0_def by auto
          ultimately show ?thesis
            using D.comp_assoc by metis
        qed
        also have "... = (map\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a]"
          unfolding map\<^sub>0_def by simp
        finally show "(map\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a]
                        = map\<^sub>1 a \<cdot>\<^sub>D (H.unit a \<star>\<^sub>D map\<^sub>0 a)"
          by simp
      qed
      show "\<And>f g. \<lbrakk>C.ide f; C.ide g; src\<^sub>C g = trg\<^sub>C f\<rbrakk> \<Longrightarrow>
                     (map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                     \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                     (map\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
                     D.inv \<a>\<^sub>D[H g, map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                     (H g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[H g, H f, map\<^sub>0 (src\<^sub>C f)]
                       = map\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f))"
      proof -
        fix f g
        assume f: "C.ide f" and g: "C.ide g" and fg: "src\<^sub>C g = trg\<^sub>C f"
        have "(map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
              \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
              (map\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
              D.inv \<a>\<^sub>D[H g, map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
              (H g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[H g, H f, map\<^sub>0 (src\<^sub>C f)]
                = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                  \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                  (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<cdot>\<^sub>D
                  (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<cdot>\<^sub>D
                   (\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g)) \<cdot>\<^sub>D
                   \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g)]
                      \<star>\<^sub>D F f) \<cdot>\<^sub>D
                  \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                  (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C f), \<sigma>\<^sub>0 (trg\<^sub>C f), F f] \<cdot>\<^sub>D
                          (\<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                          (\<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
          unfolding map\<^sub>0_def map\<^sub>1_def
          using f g fg by simp
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C f), \<sigma>\<^sub>0 (trg\<^sub>C f), F f]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
          using f g fg D.whisker_left D.whisker_right D.comp_assoc by fastforce
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (((\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C f), \<sigma>\<^sub>0 (trg\<^sub>C f), F f]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
          using f g fg D.pentagon' D.comp_assoc
                D.invert_side_of_triangle(2)
                  [of "\<a>\<^sub>D\<^sup>-\<^sup>1[H g \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f]"
                      "(\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g), F f]"
                      "H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]"]
          by force
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f] \<cdot>\<^sub>D
                         (((H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C f), \<sigma>\<^sub>0 (trg\<^sub>C f), F f])) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f)) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
         proof -
          have "((\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]
                  = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f)"
            using f g fg D.assoc'_naturality [of "\<rho>\<^sub>1 g" "\<sigma>\<^sub>0 (src\<^sub>C g)" "F f"] by simp
          thus ?thesis
             using D.comp_assoc by simp
        qed
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f)) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
       proof -
          have "((H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C f), \<sigma>\<^sub>0 (trg\<^sub>C f), F f])) \<cdot>\<^sub>D
                (H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f)
                  = H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f"
          proof -
            have "(H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                  (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f])
                    = H g \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f"
            proof -
              have "(H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                    (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f])
                      = H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                        \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (src\<^sub>C g), \<sigma>\<^sub>0 (src\<^sub>C g), F f]"
                using f g fg D.whisker_left by simp
              also have "... = H g \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f"
                using f g fg D.comp_arr_inv' by simp
              finally show ?thesis by simp
            qed
            moreover have "(H g \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                           (H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f)
                             = (H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f)"
              using f g fg D.comp_cod_arr by simp
            ultimately show ?thesis
              using fg by simp
          qed
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>1 f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
          using f g fg D.assoc'_naturality [of "H g" "\<rho>\<^sub>0 (src\<^sub>C g)" "\<sigma>\<^sub>1 f"] D.comp_assoc by simp
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 g \<star>\<^sub>D G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)])) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
        proof -
          have "(\<rho>\<^sub>1 g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f) \<cdot>\<^sub>D ((H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>1 f) =
                ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D (\<rho>\<^sub>1 g \<star>\<^sub>D G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
            using f g fg D.interchange D.comp_arr_dom D.comp_cod_arr
            by (metis C.ideD(1) \<rho>.map\<^sub>1_simps(1,5) \<rho>.naturality \<sigma>.map\<^sub>1_simps(1,4)
                \<sigma>.naturality C.ideD(2,3))
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 g \<star>\<^sub>D G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)])) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         ((H g \<star>\<^sub>D \<rho>\<^sub>1 f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
          using f g fg D.comp_assoc D.hcomp_reassoc(2) by simp
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>1 g \<star>\<^sub>D G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((H g \<star>\<^sub>D \<rho>\<^sub>1 f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
        proof -  
          have "(\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                (H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)])) \<cdot>\<^sub>D
                \<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f, \<sigma>\<^sub>0 (src\<^sub>C f)]
                  = \<a>\<^sub>D[H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                    (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
          proof -
            have "((H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                  \<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                  (\<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))
                    = \<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                      \<a>\<^sub>D[H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
              using f g fg D.pentagon D.comp_assoc by simp
            moreover have "D.seq \<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]
                                 \<a>\<^sub>D[H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
              using f g fg
              by (intro D.seqI) auto
            ultimately show ?thesis
              using f g fg D.comp_assoc
                    D.invert_opposite_sides_of_square
                      [of "\<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
                          "\<a>\<^sub>D[H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
                          "(H g \<star>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                             \<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
                          "\<a>\<^sub>D[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"]
              by fastforce
          qed
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (((\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((H g \<star>\<^sub>D \<rho>\<^sub>1 f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
        proof -
          have "(\<rho>\<^sub>1 g \<star>\<^sub>D G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D \<a>\<^sub>D[H g \<star>\<^sub>D \<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)] =
                \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D ((\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
            using f g fg D.assoc_naturality [of "\<rho>\<^sub>1 g" "G f" "\<sigma>\<^sub>0 (src\<^sub>C f)"] by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)])"
        proof -
          have "((\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                  (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                  ((H g \<star>\<^sub>D \<rho>\<^sub>1 f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) =
                (\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f] \<cdot>\<^sub>D (H g \<star>\<^sub>D \<rho>\<^sub>1 f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
            using f g fg D.whisker_right by simp
          also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<cdot>\<^sub>D
                           \<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D
                           (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)]
                             \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
          proof -
            have "((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D
                  ((\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f] \<cdot>\<^sub>D (H g \<star>\<^sub>D \<rho>\<^sub>1 f))) \<cdot>\<^sub>D
                  \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)]
                    = \<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f))"
              using f g fg \<rho>.respects_hcomp D.comp_assoc by simp
            moreover have "D.seq (\<rho>\<^sub>1 (g \<star>\<^sub>C f)) (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f))"
              using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char by auto
            ultimately
            have "(\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D
                  ((\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f] \<cdot>\<^sub>D (H g \<star>\<^sub>D \<rho>\<^sub>1 f))
                    = (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)]"
              using f g fg D.invert_side_of_triangle(2) by simp
            moreover have "D.seq (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f)))
                                 \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)]"
              using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def by auto
            ultimately
            have "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D
                  ((\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f] \<cdot>\<^sub>D (H g \<star>\<^sub>D \<rho>\<^sub>1 f))
                    = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<cdot>\<^sub>D
                      (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D
                      (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)]"
              using f g fg
                    D.invert_side_of_triangle(1)
                      [of "(\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)]"
                          "\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)"
                          "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D ((\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<cdot>\<^sub>D
                           \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f] \<cdot>\<^sub>D (H g \<star>\<^sub>D \<rho>\<^sub>1 f))"]
              by simp
            hence "(\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f] \<cdot>\<^sub>D (H g \<star>\<^sub>D \<rho>\<^sub>1 f)
                     = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<cdot>\<^sub>D
                       (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D
                       (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)]"
              using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
                    D.invert_side_of_triangle(1)
              by simp
            hence "(\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f] \<cdot>\<^sub>D (H g \<star>\<^sub>D \<rho>\<^sub>1 f)
                     = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<cdot>\<^sub>D
                       \<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D
                       (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)]"
              using D.comp_assoc by simp
            thus ?thesis
              using fg by simp
          qed
          also have "... = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                           ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                           (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                           ((\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                           (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
            using f g fg D.whisker_right C.VV.arr_char C.VV.dom_char C.VV.cod_char
                  H.FF_def G.FF_def
            by force
          finally have "((\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                        (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, \<rho>\<^sub>0 (trg\<^sub>C f), G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                        ((H g \<star>\<^sub>D \<rho>\<^sub>1 f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))
                          = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                            ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                            (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                            ((\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                            (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H g \<star>\<^sub>D H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]"
        proof -
          have "((\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)])) \<cdot>\<^sub>D
                \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]
                  = \<a>\<^sub>D\<^sup>-\<^sup>1[H g \<star>\<^sub>D H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]"
          proof -
            have "(\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                  \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                  (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)])
                    = \<a>\<^sub>D\<^sup>-\<^sup>1[H g \<star>\<^sub>D H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                      \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"
              using f g fg D.pentagon' D.comp_assoc by simp
            moreover have "D.seq (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))
                                 (\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                                   (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]))"
              using f g fg by auto
            ultimately show ?thesis
              using f g fg D.comp_assoc
                    D.invert_side_of_triangle(2)
                      [of "(\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                             \<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                             (H g \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)])"
                          "\<a>\<^sub>D\<^sup>-\<^sup>1[H g \<star>\<^sub>D H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]"
                          "\<a>\<^sub>D\<^sup>-\<^sup>1[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)]"]
              by simp
          qed
          thus ?thesis
            using D.comp_assoc by simp
        qed              
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
          using f g fg D.assoc'_naturality [of "\<Phi>\<^sub>H (g, f)" "\<rho>\<^sub>0 (src\<^sub>C f)" "\<sigma>\<^sub>0 (src\<^sub>C f)"]
          by fastforce
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D F g, F f] \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
          using f g fg D.hcomp_reassoc(1) D.comp_assoc by simp
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D F g, F f] \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g) \<star>\<^sub>D \<sigma>\<^sub>1 f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
          using f g fg D.comp_assoc D.pentagon
                D.invert_opposite_sides_of_square
                  [of "\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f]"
                      "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g)] \<star>\<^sub>D F f)"
                      "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, \<sigma>\<^sub>0 (src\<^sub>C g) \<star>\<^sub>D F f]"
                      "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f]"]
          by simp
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D F g, F f] \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g \<star>\<^sub>D \<sigma>\<^sub>1 f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
          using f g fg D.assoc_naturality [of "\<rho>\<^sub>0 (trg\<^sub>C g)" "G g" "\<sigma>\<^sub>1 f"] D.comp_assoc by simp
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D F g, F f] \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f])) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
        proof -
          have "(\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
                (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g \<star>\<^sub>D \<sigma>\<^sub>1 f)
                  = \<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D (\<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<sigma>\<^sub>1 f)"
            using f g fg D.whisker_left by simp
          also have "... = \<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                              (\<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                              \<sigma>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D
                              (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
          proof -
            have "((\<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                  \<a>\<^sub>D[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                  (\<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<sigma>\<^sub>1 f)) \<cdot>\<^sub>D
                  \<a>\<^sub>D[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]
                    = \<sigma>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
              using f g fg \<sigma>.respects_hcomp D.comp_assoc by simp
            hence "(\<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                   (\<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<sigma>\<^sub>1 f)
                     = (\<sigma>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
              using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char D.invert_side_of_triangle(2)
              by simp
            hence "\<a>\<^sub>D[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                   (\<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<sigma>\<^sub>1 f)
                     = (\<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                       (\<sigma>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
              using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char G.FF_def
                    D.invert_side_of_triangle(1)
                      [of "(\<sigma>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
                          "\<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)"
                          "\<a>\<^sub>D[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D (\<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
                             \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<sigma>\<^sub>1 f)"]
              by simp
            hence "(\<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<sigma>\<^sub>1 f)
                     = \<a>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D (\<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                       (\<sigma>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
              using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char G.FF_def F.FF_def
                    D.invert_side_of_triangle(1)
              by simp
            thus ?thesis
              using D.comp_assoc by simp
          qed
          also have "... = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f]) \<cdot>\<^sub>D
                           (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                           (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                           (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                           (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)])"
             using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char F.FF_def G.FF_def
                   D.whisker_left
            by force
          finally have "(\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
                        (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f]) \<cdot>\<^sub>D
                        (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g \<star>\<^sub>D \<sigma>\<^sub>1 f)
                          = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f]) \<cdot>\<^sub>D
                            (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                            (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                            (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                            (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)])"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g \<star>\<^sub>D F f] \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f)))) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
          using f g fg D.pentagon' D.comp_assoc
                D.invert_side_of_triangle(1)
                  [of "(\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g] \<star>\<^sub>D F f) \<cdot>\<^sub>D
                       \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D F g, F f] \<cdot>\<^sub>D
                       (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f])"
                      "\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f]"
                      "\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F g \<star>\<^sub>D F f]" ]
          by simp
        also have "... = ((((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f)))) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
            using f g fg D.assoc'_naturality [of "\<rho>\<^sub>0 (trg\<^sub>C g)" "\<sigma>\<^sub>0 (trg\<^sub>C g)" "D.inv (\<Phi>\<^sub>F (g, f))"]
                  D.comp_assoc
            by simp
        also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
        proof -
          have "(((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f)))) \<cdot>\<^sub>D
                \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)]
                  = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)]"
          proof -
            have "((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                  ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f)))
                    = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f))"
              using f g fg D.whisker_left C.VV.arr_char by simp
            also have "... = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D F (g \<star>\<^sub>C f)"
              using f g fg D.comp_arr_inv' F.cmp_components_are_iso by simp
            finally have "((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                          ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D D.inv (\<Phi>\<^sub>F (g, f)))
                            = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D F (g \<star>\<^sub>C f)"
              by blast
            moreover have "((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C g)) \<star>\<^sub>D F (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                           \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)]
                             = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)]"
              using f g fg D.comp_cod_arr by simp
            ultimately show ?thesis by simp
          qed
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g \<star>\<^sub>D G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
          using f g fg D.pentagon D.comp_assoc
                D.invert_side_of_triangle(1)
                  [of "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
                      "\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
                      "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g \<star>\<^sub>D G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                (\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"]
                D.invert_side_of_triangle(2)
                  [of "(\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
                      "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g \<star>\<^sub>D G f, \<sigma>\<^sub>0 (src\<^sub>C f)]"
                      "\<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"]
          by simp
        also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G (g \<star>\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         ((((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
         proof -
          have "(\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g \<star>\<^sub>D G f, \<sigma>\<^sub>0 (src\<^sub>C f)]
                  = \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G (g \<star>\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                    ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
            using f g fg D.assoc_naturality [of "\<rho>\<^sub>0 (trg\<^sub>C g)" "\<Phi>\<^sub>G (g, f)" "\<sigma>\<^sub>0 (src\<^sub>C f)"]
            by fastforce
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C g), \<sigma>\<^sub>0 (trg\<^sub>C g), F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D
                         (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<sigma>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G (g \<star>\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[H (g \<star>\<^sub>C f), \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                         (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
        proof -
          have "(((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))) \<cdot>\<^sub>D
                (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))
                  = \<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
          proof -
            have "((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                  ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))
                    = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                      (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)))
                         \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
              using f g fg C.VV.arr_char D.whisker_right by simp
            also have "... = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
              using f g fg C.VV.arr_char D.whisker_left [of "\<rho>\<^sub>0 (trg\<^sub>C g)"] by simp
            also have "... = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G (g \<star>\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
              using f g fg D.comp_arr_inv' G.cmp_components_are_iso G.cmp_simps(5) by auto
            finally have "((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                          ((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))
                            = (\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G (g \<star>\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
              by blast
            moreover have "((\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D G (g \<star>\<^sub>C f)) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                           (\<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))
                             = \<rho>\<^sub>1 (g \<star>\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
              using f g fg D.comp_cod_arr by simp
            ultimately show ?thesis by simp
          qed
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = map\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f))"
          unfolding map\<^sub>0_def map\<^sub>1_def
          using f g fg D.comp_assoc by simp
        finally show "(map\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                      \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D
                      (map\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D
                      D.inv \<a>\<^sub>D[H g, map\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D
                      (H g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D
                      \<a>\<^sub>D[H g, H f, map\<^sub>0 (src\<^sub>C f)]
                        = map\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>C f))"
          using D.comp_assoc by simp
      qed
    qed
 
    lemma is_pseudonatural_transformation:
    shows "pseudonatural_transformation
             V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F H \<Phi>\<^sub>H map\<^sub>0 map\<^sub>1"
      ..
 
   end
 
   subsection "Whiskering of Pseudonatural Transformations"
 
   text \<open>
     Similarly to ordinary natural transformations, pseudonatural transformations can be whiskered
     with pseudofunctors on the left and the right.
   \<close>
 
   locale pseudonatural_transformation_whisker_right =
     B: bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B +
     C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C +
     D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D +
     \<tau>.F: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F +
     \<tau>.G: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G +
     H: pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C H \<Phi>\<^sub>H +
     \<tau>: pseudonatural_transformation
          V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
   for V\<^sub>B :: "'b comp"                   (infixr "\<cdot>\<^sub>B" 55)
   and H\<^sub>B :: "'b comp"                   (infixr "\<star>\<^sub>B" 53)
   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"       ("\<a>\<^sub>B[_, _, _]")
   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"                   ("\<i>\<^sub>B[_]")
   and src\<^sub>B :: "'b \<Rightarrow> 'b"
   and trg\<^sub>B :: "'b \<Rightarrow> 'b"
   and V\<^sub>C :: "'c comp"                   (infixr "\<cdot>\<^sub>C" 55)
   and H\<^sub>C :: "'c comp"                   (infixr "\<star>\<^sub>C" 53)
   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"       ("\<a>\<^sub>C[_, _, _]")
   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"                   ("\<i>\<^sub>C[_]")
   and src\<^sub>C :: "'c \<Rightarrow> 'c"
   and trg\<^sub>C :: "'c \<Rightarrow> 'c"
   and V\<^sub>D :: "'d comp"                   (infixr "\<cdot>\<^sub>D" 55)
   and H\<^sub>D :: "'d comp"                   (infixr "\<star>\<^sub>D" 53)
   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"       ("\<a>\<^sub>D[_, _, _]")
   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"                   ("\<i>\<^sub>D[_]")
   and src\<^sub>D :: "'d \<Rightarrow> 'd"
   and trg\<^sub>D :: "'d \<Rightarrow> 'd"
   and F :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>F :: "'c * 'c \<Rightarrow> 'd"
   and G :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>G :: "'c * 'c \<Rightarrow> 'd"
   and H :: "'b \<Rightarrow> 'c"
   and \<Phi>\<^sub>H :: "'b * 'b \<Rightarrow> 'c"
   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"
   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd"
   begin
 
     interpretation FoH: composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                           V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H F \<Phi>\<^sub>F
       ..
     interpretation GoH: composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                           V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H G \<Phi>\<^sub>G
       ..
 
     definition map\<^sub>0
     where "map\<^sub>0 a = \<tau>\<^sub>0 (H.map\<^sub>0 a)"
 
     definition map\<^sub>1
     where "map\<^sub>1 f = \<tau>\<^sub>1 (H f)"
 
     sublocale pseudonatural_transformation V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                           \<open>F o H\<close> FoH.cmp \<open>G o H\<close> GoH.cmp map\<^sub>0 map\<^sub>1
     proof
       show "\<And>a. B.obj a \<Longrightarrow> D.ide (map\<^sub>0 a)"
         using map\<^sub>0_def by simp
       show "\<And>a. B.obj a \<Longrightarrow> \<guillemotleft>map\<^sub>0 a : src\<^sub>D ((F \<circ> H) a) \<rightarrow>\<^sub>D src\<^sub>D ((G \<circ> H) a)\<guillemotright>"
         using map\<^sub>0_def \<tau>.map\<^sub>0_in_hhom B.obj_simps C.obj_simps by simp
       show "\<And>f. B.ide f \<Longrightarrow> D.iso (map\<^sub>1 f)"
         using map\<^sub>1_def \<tau>.iso_map\<^sub>1_ide by simp
       show "\<And>f. B.ide f \<Longrightarrow>
                    \<guillemotleft>map\<^sub>1 f : (G \<circ> H) f \<star>\<^sub>D map\<^sub>0 (src\<^sub>B f) \<Rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>B f) \<star>\<^sub>D (F \<circ> H) f\<guillemotright>"
         using map\<^sub>0_def map\<^sub>1_def by auto
       show "\<And>\<mu>. B.arr \<mu> \<Longrightarrow>
                    map\<^sub>1 (B.cod \<mu>) \<cdot>\<^sub>D ((G \<circ> H) \<mu> \<star>\<^sub>D map\<^sub>0 (src\<^sub>B \<mu>)) =
                    (map\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>D (F \<circ> H) \<mu>) \<cdot>\<^sub>D map\<^sub>1 (B.dom \<mu>)"
         unfolding map\<^sub>0_def map\<^sub>1_def
         using \<tau>.naturality by force
       show "\<And>a. B.obj a \<Longrightarrow>
                   (map\<^sub>0 a \<star>\<^sub>D FoH.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a]
                      = map\<^sub>1 a \<cdot>\<^sub>D (GoH.unit a \<star>\<^sub>D map\<^sub>0 a)"
       proof -
         fix a
         assume a: "B.obj a"
         have "map\<^sub>1 a \<cdot>\<^sub>D (GoH.unit a \<star>\<^sub>D map\<^sub>0 a) =
               \<tau>\<^sub>1 (H a) \<cdot>\<^sub>D (G (H.unit a) \<cdot>\<^sub>D \<tau>.G.unit (H.map\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0 (H.map\<^sub>0 a))"
           unfolding map\<^sub>0_def map\<^sub>1_def
           using a GoH.unit_char' by simp
         also have "... = (\<tau>\<^sub>1 (H a) \<cdot>\<^sub>D (G (H.unit a) \<star>\<^sub>D \<tau>\<^sub>0 (H.map\<^sub>0 a))) \<cdot>\<^sub>D
                          (\<tau>.G.unit (H.map\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0 (H.map\<^sub>0 a))"
           using a D.whisker_right D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0 (H.map\<^sub>0 a) \<star>\<^sub>D F (H.unit a)) \<cdot>\<^sub>D \<tau>\<^sub>1 (H.map\<^sub>0 a) \<cdot>\<^sub>D
                          (\<tau>.G.unit (H.map\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0 (H.map\<^sub>0 a))"
           using a \<tau>.naturality [of "H.unit a"] D.comp_assoc by simp
         also have "... = ((\<tau>\<^sub>0 (H.map\<^sub>0 a) \<star>\<^sub>D F (H.unit a)) \<cdot>\<^sub>D
                           (\<tau>\<^sub>0 (H.map\<^sub>0 a) \<star>\<^sub>D \<tau>.F.unit (H.map\<^sub>0 a))) \<cdot>\<^sub>D
                          \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 (H.map\<^sub>0 a)] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0 (H.map\<^sub>0 a)]"
           using a \<tau>.respects_unit D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0 (H.map\<^sub>0 a) \<star>\<^sub>D F (H.unit a) \<cdot>\<^sub>D \<tau>.F.unit (H.map\<^sub>0 a)) \<cdot>\<^sub>D
                          \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 (H.map\<^sub>0 a)] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0 (H.map\<^sub>0 a)]"
           using a D.whisker_left by simp
         also have "... = (map\<^sub>0 a \<star>\<^sub>D FoH.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a]"
           unfolding map\<^sub>0_def map\<^sub>1_def
           using a FoH.unit_char' by simp
         finally show "(map\<^sub>0 a \<star>\<^sub>D FoH.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a]
                         = map\<^sub>1 a \<cdot>\<^sub>D (GoH.unit a \<star>\<^sub>D map\<^sub>0 a)"
           by simp
       qed
       show "\<And>f g. \<lbrakk>B.ide f; B.ide g; src\<^sub>B g = trg\<^sub>B f\<rbrakk> \<Longrightarrow>
                     (map\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>D FoH.cmp (g, f)) \<cdot>\<^sub>D
                     \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>B g), (F \<circ> H) g, (F \<circ> H) f] \<cdot>\<^sub>D
                     (map\<^sub>1 g \<star>\<^sub>D (F \<circ> H) f) \<cdot>\<^sub>D
                     D.inv \<a>\<^sub>D[(G \<circ> H) g, map\<^sub>0 (src\<^sub>B g), (F \<circ> H) f] \<cdot>\<^sub>D
                     ((G \<circ> H) g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D
                     \<a>\<^sub>D[(G \<circ> H) g, (G \<circ> H) f, map\<^sub>0 (src\<^sub>B f)]
                       = map\<^sub>1 (g \<star>\<^sub>B f) \<cdot>\<^sub>D (GoH.cmp (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>B f))"
       proof -
         fix f g
         assume f: "B.ide f" and g: "B.ide g" and fg: "src\<^sub>B g = trg\<^sub>B f"
         have "map\<^sub>1 (g \<star>\<^sub>B f) \<cdot>\<^sub>D (GoH.cmp (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>B f))
                 = \<tau>\<^sub>1 (H (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                   (G (H (g \<star>\<^sub>B f)) \<cdot>\<^sub>D G (\<Phi>\<^sub>H (g, f)) \<cdot>\<^sub>D \<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))"
           unfolding map\<^sub>0_def map\<^sub>1_def
           using f g fg GoH.cmp_def B.VV.arr_char B.VV.dom_char by simp
         also have "... = (\<tau>\<^sub>1 (H (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                          (G (H (g \<star>\<^sub>B f)) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))) \<cdot>\<^sub>D
                          (G (\<Phi>\<^sub>H (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))"
           using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char H.FF_def
                 C.VV.arr_char C.VV.dom_char C.VV.cod_char D.whisker_right
                 D.comp_assoc
           by simp
         also have "... = (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D F (H (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          (\<tau>\<^sub>1 (H (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                          (G (\<Phi>\<^sub>H (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))"
           using f g fg \<tau>.naturality [of "H (g \<star>\<^sub>B f)"] D.comp_assoc by auto
         also have "... = (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D F (H (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D F (\<Phi>\<^sub>H (g, f))) \<cdot>\<^sub>D
                          \<tau>\<^sub>1 (H g \<star>\<^sub>C H f) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))"
           using f g fg \<tau>.naturality [of "\<Phi>\<^sub>H (g, f)"] D.comp_assoc by fastforce
         also have "... = (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D F (H (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D F (\<Phi>\<^sub>H (g, f))) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D \<Phi>\<^sub>F (H g, H f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C (H g)), F (H g), F (H f)] \<cdot>\<^sub>D
                          (\<tau>\<^sub>1 (H g) \<star>\<^sub>D F (H f)) \<cdot>\<^sub>D
                          D.inv \<a>\<^sub>D[G (H g), \<tau>\<^sub>0 (src\<^sub>C (H g)), F (H f)] \<cdot>\<^sub>D
                          (G (H g) \<star>\<^sub>D \<tau>\<^sub>1 (H f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))] \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))))"
         proof -
           have "\<tau>\<^sub>1 (H g \<star>\<^sub>C H f) = (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D \<Phi>\<^sub>F (H g, H f)) \<cdot>\<^sub>D
                                  \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C (H g)), F (H g), F (H f)] \<cdot>\<^sub>D
                                  (\<tau>\<^sub>1 (H g) \<star>\<^sub>D F (H f)) \<cdot>\<^sub>D
                                  D.inv \<a>\<^sub>D[G (H g), \<tau>\<^sub>0 (src\<^sub>C (H g)), F (H f)] \<cdot>\<^sub>D
                                  (G (H g) \<star>\<^sub>D \<tau>\<^sub>1 (H f)) \<cdot>\<^sub>D
                                  \<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))] \<cdot>\<^sub>D
                                  D.inv (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))"
           proof -
             have "\<tau>\<^sub>1 (H g \<star>\<^sub>C H f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))
                     = (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D \<Phi>\<^sub>F (H g, H f)) \<cdot>\<^sub>D
                       \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C (H g)), F (H g), F (H f)] \<cdot>\<^sub>D
                       (\<tau>\<^sub>1 (H g) \<star>\<^sub>D F (H f)) \<cdot>\<^sub>D
                       D.inv \<a>\<^sub>D[G (H g), \<tau>\<^sub>0 (src\<^sub>C (H g)), F (H f)] \<cdot>\<^sub>D
                       (G (H g) \<star>\<^sub>D \<tau>\<^sub>1 (H f)) \<cdot>\<^sub>D
                       \<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))]"
               using f g fg \<tau>.respects_hcomp [of "H f" "H g"] by simp
             moreover have "D.seq (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D \<Phi>\<^sub>F (H g, H f))
                                  (\<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C (H g)), F (H g), F (H f)] \<cdot>\<^sub>D
                                  (\<tau>\<^sub>1 (H g) \<star>\<^sub>D F (H f)) \<cdot>\<^sub>D
                                  D.inv \<a>\<^sub>D[G (H g), \<tau>\<^sub>0 (src\<^sub>C (H g)), F (H f)] \<cdot>\<^sub>D
                                  (G (H g) \<star>\<^sub>D \<tau>\<^sub>1 (H f)) \<cdot>\<^sub>D
                                  \<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))])"
               using f g fg C.obj_simps C.VV.arr_char C.VV.dom_char C.VV.cod_char \<tau>.F.FF_def
               by simp
             moreover have "D.iso (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))"
               using f g fg \<tau>.G.cmp_components_are_iso [of "H g" "H f"] by simp
             ultimately
             have "\<tau>\<^sub>1 (H g \<star>\<^sub>C H f) = ((\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D \<Phi>\<^sub>F (H g, H f)) \<cdot>\<^sub>D
                                    \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C (H g)), F (H g), F (H f)] \<cdot>\<^sub>D
                                    (\<tau>\<^sub>1 (H g) \<star>\<^sub>D F (H f)) \<cdot>\<^sub>D
                                    D.inv \<a>\<^sub>D[G (H g), \<tau>\<^sub>0 (src\<^sub>C (H g)), F (H f)] \<cdot>\<^sub>D
                                    (G (H g) \<star>\<^sub>D \<tau>\<^sub>1 (H f)) \<cdot>\<^sub>D
                                    \<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))]) \<cdot>\<^sub>D
                                    D.inv (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))"
             using D.invert_side_of_triangle(2) by blast
             thus ?thesis
               using D.comp_assoc by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D F (H (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D F (\<Phi>\<^sub>H (g, f))) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D \<Phi>\<^sub>F (H g, H f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C (H g)), F (H g), F (H f)] \<cdot>\<^sub>D
                          (\<tau>\<^sub>1 (H g) \<star>\<^sub>D F (H f)) \<cdot>\<^sub>D
                          D.inv \<a>\<^sub>D[G (H g), \<tau>\<^sub>0 (src\<^sub>C (H g)), F (H f)] \<cdot>\<^sub>D
                          (G (H g) \<star>\<^sub>D \<tau>\<^sub>1 (H f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))]"
         proof -
           have "(D.inv (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))) \<cdot>\<^sub>D
                 (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))))
                   = (G (H g) \<star>\<^sub>D G (H f)) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))"
           proof -
             have "(D.inv (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))) \<cdot>\<^sub>D
                   (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))))
                     = (D.inv (\<Phi>\<^sub>G (H g, H f)) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))) \<cdot>\<^sub>D
                       (\<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f)))"
               using f g fg by simp
             also have "... = D.inv (\<Phi>\<^sub>G (H g, H f)) \<cdot>\<^sub>D \<Phi>\<^sub>G (H g, H f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))"
               using f g fg D.whisker_right C.VV.arr_char by simp
             also have "... = (G (H g) \<star>\<^sub>D G (H f)) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C (H f))"
               using f g fg D.comp_inv_arr' \<tau>.G.cmp_components_are_iso by simp
             finally show ?thesis by blast
           qed
           moreover have "\<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))] \<cdot>\<^sub>D ...
                            = \<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))]"
             using f g fg D.comp_arr_dom by simp
           ultimately show ?thesis by simp
         qed
         also have "... = (\<tau>\<^sub>0 (trg\<^sub>C (H g)) \<star>\<^sub>D F (H (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                                             F (\<Phi>\<^sub>H (g, f)) \<cdot>\<^sub>D
                                             \<Phi>\<^sub>F (H g, H f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C (H g)), F (H g), F (H f)] \<cdot>\<^sub>D
                          (\<tau>\<^sub>1 (H g) \<star>\<^sub>D F (H f)) \<cdot>\<^sub>D
                          D.inv \<a>\<^sub>D[G (H g), \<tau>\<^sub>0 (src\<^sub>C (H g)), F (H f)] \<cdot>\<^sub>D
                          (G (H g) \<star>\<^sub>D \<tau>\<^sub>1 (H f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[G (H g), G (H f), \<tau>\<^sub>0 (src\<^sub>C (H f))]"
           using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char
                 B.VV.arr_char B.VV.dom_char B.VV.cod_char H.FF_def
                 D.whisker_left C.VV.arr_char B.VV.arr_char D.comp_assoc
           by auto
         also have "... = (map\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>D FoH.cmp (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>B g), (F \<circ> H) g, (F \<circ> H) f] \<cdot>\<^sub>D
                          (map\<^sub>1 g \<star>\<^sub>D (F \<circ> H) f) \<cdot>\<^sub>D
                          D.inv \<a>\<^sub>D[(G \<circ> H) g, map\<^sub>0 (src\<^sub>B g), (F \<circ> H) f] \<cdot>\<^sub>D
                          ((G \<circ> H) g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D
                          \<a>\<^sub>D[(G \<circ> H) g, (G \<circ> H) f, map\<^sub>0 (src\<^sub>B f)]"
           unfolding map\<^sub>0_def map\<^sub>1_def
           using f g fg FoH.cmp_def B.VV.arr_char B.VV.dom_char by simp
         finally show "(map\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>D FoH.cmp (g, f)) \<cdot>\<^sub>D
                       \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>B g), (F \<circ> H) g, (F \<circ> H) f] \<cdot>\<^sub>D
                       (map\<^sub>1 g \<star>\<^sub>D (F \<circ> H) f) \<cdot>\<^sub>D
                       D.inv \<a>\<^sub>D[(G \<circ> H) g, map\<^sub>0 (src\<^sub>B g), (F \<circ> H) f] \<cdot>\<^sub>D
                       ((G \<circ> H) g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D
                       \<a>\<^sub>D[(G \<circ> H) g, (G \<circ> H) f, map\<^sub>0 (src\<^sub>B f)]
                         = map\<^sub>1 (g \<star>\<^sub>B f) \<cdot>\<^sub>D (GoH.cmp (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>B f))"
           by simp
       qed
     qed
 
     lemma is_pseudonatural_transformation:
     shows "pseudonatural_transformation V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                           (F o H) FoH.cmp (G o H) GoH.cmp map\<^sub>0 map\<^sub>1"
       ..
 
   end
 
   locale pseudonatural_transformation_whisker_left =
     B: bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B +
     C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C +
     D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D +
     \<tau>.F: pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F +
     \<tau>.G: pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G +
     H: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H +
     \<tau>: pseudonatural_transformation
          V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
   for V\<^sub>B :: "'b comp"                   (infixr "\<cdot>\<^sub>B" 55)
   and H\<^sub>B :: "'b comp"                   (infixr "\<star>\<^sub>B" 53)
   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"       ("\<a>\<^sub>B[_, _, _]")
   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"                   ("\<i>\<^sub>B[_]")
   and src\<^sub>B :: "'b \<Rightarrow> 'b"
   and trg\<^sub>B :: "'b \<Rightarrow> 'b"
   and V\<^sub>C :: "'c comp"                   (infixr "\<cdot>\<^sub>C" 55)
   and H\<^sub>C :: "'c comp"                   (infixr "\<star>\<^sub>C" 53)
   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"       ("\<a>\<^sub>C[_, _, _]")
   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"                   ("\<i>\<^sub>C[_]")
   and src\<^sub>C :: "'c \<Rightarrow> 'c"
   and trg\<^sub>C :: "'c \<Rightarrow> 'c"
   and V\<^sub>D :: "'d comp"                   (infixr "\<cdot>\<^sub>D" 55)
   and H\<^sub>D :: "'d comp"                   (infixr "\<star>\<^sub>D" 53)
   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"       ("\<a>\<^sub>D[_, _, _]")
   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"                   ("\<i>\<^sub>D[_]")
   and src\<^sub>D :: "'d \<Rightarrow> 'd"
   and trg\<^sub>D :: "'d \<Rightarrow> 'd"
   and F :: "'b \<Rightarrow> 'c"
   and \<Phi>\<^sub>F :: "'b * 'b \<Rightarrow> 'c"
   and G :: "'b \<Rightarrow> 'c"
   and \<Phi>\<^sub>G :: "'b * 'b \<Rightarrow> 'c"
   and H :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>H :: "'c * 'c \<Rightarrow> 'd"
   and \<tau>\<^sub>0 :: "'b \<Rightarrow> 'c"
   and \<tau>\<^sub>1 :: "'b \<Rightarrow> 'c"
   begin
 
     interpretation HoF: composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                           V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F H \<Phi>\<^sub>H
       ..
     interpretation HoG: composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                           V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G H \<Phi>\<^sub>H
       ..
 
     definition map\<^sub>0
     where "map\<^sub>0 a = H (\<tau>\<^sub>0 a)"
 
     definition map\<^sub>1
     where "map\<^sub>1 f = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B f), F f)) \<cdot>\<^sub>D H (\<tau>\<^sub>1 f) \<cdot>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f))"
 
     sublocale pseudonatural_transformation V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                 HoF.map HoF.cmp HoG.map HoG.cmp map\<^sub>0 map\<^sub>1
     proof
       show "\<And>a. B.obj a \<Longrightarrow> D.ide (map\<^sub>0 a)"
         using map\<^sub>0_def by simp
       show "\<And>f. B.ide f \<Longrightarrow> D.iso (map\<^sub>1 f)"
       proof -
         fix f
         assume f: "B.ide f"
         have "D.seq (H (\<tau>\<^sub>1 f)) (\<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f)))"
           using f by (intro D.seqI) auto
         moreover have "D.seq (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B f), F f)))
                              (H (\<tau>\<^sub>1 f) \<cdot>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f)))"
           using f \<tau>.map\<^sub>1_in_hom [of f] calculation by (intro D.seqI) auto
         ultimately show "D.iso (map\<^sub>1 f)"
           using f map\<^sub>1_def H.preserves_iso \<tau>.iso_map\<^sub>1_ide H.cmp_components_are_iso
                 C.VV.arr_char D.isos_compose
           by auto
       qed
       show "\<And>a. B.obj a \<Longrightarrow> \<guillemotleft>map\<^sub>0 a : src\<^sub>D ((H \<circ> F) a) \<rightarrow>\<^sub>D src\<^sub>D ((H \<circ> G) a)\<guillemotright>"
         using map\<^sub>0_def by fastforce
       show "\<And>f. B.ide f \<Longrightarrow>
                    \<guillemotleft>map\<^sub>1 f : (H \<circ> G) f \<star>\<^sub>D map\<^sub>0 (src\<^sub>B f) \<Rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>B f) \<star>\<^sub>D (H \<circ> F) f\<guillemotright>"
         using map\<^sub>0_def map\<^sub>1_def by fastforce
       show "\<And>\<mu>. B.arr \<mu> \<Longrightarrow>
                    map\<^sub>1 (B.cod \<mu>) \<cdot>\<^sub>D ((H \<circ> G) \<mu> \<star>\<^sub>D map\<^sub>0 (src\<^sub>B \<mu>))
                      = (map\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>D (H \<circ> F) \<mu>) \<cdot>\<^sub>D map\<^sub>1 (B.dom \<mu>)"
       proof -
         fix \<mu>
         assume \<mu>: "B.arr \<mu>"
         have "(map\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>D (H \<circ> F) \<mu>) \<cdot>\<^sub>D map\<^sub>1 (B.dom \<mu>)
                 = (H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>)) \<star>\<^sub>D (H \<circ> F) \<mu>) \<cdot>\<^sub>D
                   D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B (B.dom \<mu>)), F (B.dom \<mu>))) \<cdot>\<^sub>D
                   H (\<tau>\<^sub>1 (B.dom \<mu>)) \<cdot>\<^sub>D \<Phi>\<^sub>H (G (B.dom \<mu>), \<tau>\<^sub>0 (src\<^sub>B (B.dom \<mu>)))"
           unfolding map\<^sub>0_def map\<^sub>1_def
           using \<mu> by simp
         also have "... = ((H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>)) \<star>\<^sub>D H (F \<mu>)) \<cdot>\<^sub>D
                          D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.dom \<mu>)))) \<cdot>\<^sub>D
                 H (\<tau>\<^sub>1 (B.dom \<mu>)) \<cdot>\<^sub>D \<Phi>\<^sub>H (G (B.dom \<mu>), \<tau>\<^sub>0 (src\<^sub>B \<mu>))"
           using \<mu> D.comp_assoc by simp
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>C F \<mu>) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 (B.dom \<mu>))) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G (B.dom \<mu>), \<tau>\<^sub>0 (src\<^sub>B (B.dom \<mu>)))"
         proof -
           have "\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>)) \<cdot>\<^sub>D (H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>)) \<star>\<^sub>D H (F \<mu>))
                   = H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>C F \<mu>) \<cdot>\<^sub>D \<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.dom \<mu>))"
             using \<mu> H.\<Phi>.naturality [of "(\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F \<mu>)"]
                   C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
             by simp
           moreover have "D.seq (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>))) (H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>)) \<star>\<^sub>D H (F \<mu>))"
             using \<mu>
             by (intro D.seqI D.hseqI') auto
           moreover have "D.iso (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>)))"
             using \<mu> H.cmp_components_are_iso by simp
           moreover have "D.iso (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.dom \<mu>)))"
             using \<mu> H.cmp_components_are_iso by simp
           ultimately
           have "(H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>)) \<star>\<^sub>D H (F \<mu>)) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.dom \<mu>)))
                 = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>))) \<cdot>\<^sub>D H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>C F \<mu>)"
             using \<mu> H.cmp_components_are_iso C.VV.arr_char D.invert_opposite_sides_of_square
             by blast
           thus ?thesis
             using \<mu> D.comp_assoc by simp
         qed
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>))) \<cdot>\<^sub>D
                          H ((\<tau>\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>C F \<mu>) \<cdot>\<^sub>C \<tau>\<^sub>1 (B.dom \<mu>)) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G (B.dom \<mu>), \<tau>\<^sub>0 (src\<^sub>B (B.dom \<mu>)))"
           using \<mu> by simp
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>))) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 (B.cod \<mu>) \<cdot>\<^sub>C (G \<mu> \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B \<mu>))) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G (B.dom \<mu>), \<tau>\<^sub>0 (src\<^sub>B (B.dom \<mu>)))"
           using \<mu> \<tau>.naturality by simp
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>))) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 (B.cod \<mu>)) \<cdot>\<^sub>D
                          H (G \<mu> \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B \<mu>)) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G (B.dom \<mu>), \<tau>\<^sub>0 (src\<^sub>B (B.dom \<mu>)))"
           using \<mu> D.comp_assoc by simp
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B \<mu>), F (B.cod \<mu>))) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 (B.cod \<mu>)) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G (B.cod \<mu>), \<tau>\<^sub>0 (src\<^sub>B (B.cod \<mu>))) \<cdot>\<^sub>D
                          (H (G \<mu>) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B \<mu>)))"
           using \<mu> H.\<Phi>.naturality [of "(G \<mu>, \<tau>\<^sub>0 (src\<^sub>B \<mu>))"]
                 C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
           by force
         also have "... = map\<^sub>1 (B.cod \<mu>) \<cdot>\<^sub>D ((H \<circ> G) \<mu> \<star>\<^sub>D map\<^sub>0 (src\<^sub>B \<mu>))"
           unfolding map\<^sub>0_def map\<^sub>1_def
           using \<mu> D.comp_assoc by simp
         finally show "map\<^sub>1 (B.cod \<mu>) \<cdot>\<^sub>D ((H \<circ> G) \<mu> \<star>\<^sub>D map\<^sub>0 (src\<^sub>B \<mu>))
                         = (map\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>D (H \<circ> F) \<mu>) \<cdot>\<^sub>D map\<^sub>1 (B.dom \<mu>)"
           by simp
       qed
       show "\<And>a. B.obj a \<Longrightarrow> (map\<^sub>0 a \<star>\<^sub>D HoF.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a]
                                = map\<^sub>1 a \<cdot>\<^sub>D (HoG.unit a \<star>\<^sub>D map\<^sub>0 a)"
       proof -
         fix a
         assume a: "B.obj a"
         have "map\<^sub>1 a \<cdot>\<^sub>D (HoG.unit a \<star>\<^sub>D map\<^sub>0 a)
                 = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)) \<cdot>\<^sub>D
                   H (\<tau>\<^sub>1 a) \<cdot>\<^sub>D
                   \<Phi>\<^sub>H (G a, \<tau>\<^sub>0 a) \<cdot>\<^sub>D
                   (H (\<tau>.G.unit a) \<cdot>\<^sub>D H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
           unfolding map\<^sub>0_def map\<^sub>1_def
           using a HoG.unit_char' D.comp_assoc by auto
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 a) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G a, \<tau>\<^sub>0 a) \<cdot>\<^sub>D
                          (H (\<tau>.G.unit a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))) \<cdot>\<^sub>D
                          (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
           using a D.whisker_right D.comp_assoc by simp
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>1 a) \<cdot>\<^sub>D
                          H (\<tau>.G.unit a \<star>\<^sub>C \<tau>\<^sub>0 a)) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a) \<cdot>\<^sub>D
                          (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
           using a H.\<Phi>.naturality [of "(\<tau>.G.unit a, \<tau>\<^sub>0 a)"] D.comp_assoc
                 C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
           by auto
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 a \<cdot>\<^sub>C (\<tau>.G.unit a \<star>\<^sub>C \<tau>\<^sub>0 a)) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a) \<cdot>\<^sub>D
                          (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
         proof -
           have "C.arr (\<tau>\<^sub>1 a \<cdot>\<^sub>C (\<tau>.G.unit a \<star>\<^sub>C \<tau>\<^sub>0 a))"
             using a by force
          hence "H (\<tau>\<^sub>1 a) \<cdot>\<^sub>D H (\<tau>.G.unit a \<star>\<^sub>C \<tau>\<^sub>0 a) = H (\<tau>\<^sub>1 a \<cdot>\<^sub>C (\<tau>.G.unit a \<star>\<^sub>C \<tau>\<^sub>0 a))"
             using a by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)) \<cdot>\<^sub>D
                          H ((\<tau>\<^sub>0 a \<star>\<^sub>C \<tau>.F.unit a) \<cdot>\<^sub>C \<r>\<^sub>C\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>C \<l>\<^sub>C[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a) \<cdot>\<^sub>D
                          (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
           using a \<tau>.respects_unit by simp
         also have "... = (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>0 a \<star>\<^sub>C \<tau>.F.unit a)) \<cdot>\<^sub>D
                          H (\<r>\<^sub>C\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D
                          H (\<l>\<^sub>C[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a) \<cdot>\<^sub>D
                          (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
           using a D.comp_assoc B.obj_simps by simp
         also have "... = (H (\<tau>\<^sub>0 a) \<star>\<^sub>D H (\<tau>.F.unit a)) \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, \<tau>.F.map\<^sub>0 a)) \<cdot>\<^sub>D
                          H (\<r>\<^sub>C\<^sup>-\<^sup>1[\<tau>\<^sub>0 a])) \<cdot>\<^sub>D
                          H (\<l>\<^sub>C[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a) \<cdot>\<^sub>D
                          (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
         proof -
           have "D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)) \<cdot>\<^sub>D H (\<tau>\<^sub>0 a \<star>\<^sub>C \<tau>.F.unit a)
                   = (H (\<tau>\<^sub>0 a) \<star>\<^sub>D H (\<tau>.F.unit a)) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, \<tau>.F.map\<^sub>0 a))"
           proof -
             have "D.seq (H (\<tau>\<^sub>0 a \<star>\<^sub>C \<tau>.F.unit a)) (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, \<tau>.F.map\<^sub>0 a))"
               using a C.VV.arr_char C.VV.dom_char C.VV.cod_char C.obj_simps by auto
             moreover have "D.iso (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)) \<and> D.iso (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, \<tau>.F.map\<^sub>0 a))"
             proof -
               have "C.ide (F a) \<and> src\<^sub>C (\<tau>\<^sub>0 a) = trg\<^sub>C (F a) \<and> src\<^sub>C (\<tau>\<^sub>0 a) = trg\<^sub>C (\<tau>.F.map\<^sub>0 a)"
                 using a by auto
               moreover have "C.ide (\<tau>.F.map\<^sub>0 a)"
               proof -
                 (* TODO: I still haven't been able to configure the simps to do this. *)
                 have "C.obj (\<tau>.F.map\<^sub>0 a)"
                   using a by simp
                 thus ?thesis by auto
               qed
               ultimately show ?thesis
                 using a H.cmp_components_are_iso B.obj_simps by auto
             qed
             ultimately show ?thesis
               using a H.\<Phi>.naturality [of "(\<tau>\<^sub>0 a, \<tau>.F.unit a)"]
                     C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
                     D.invert_opposite_sides_of_square
                       [of "\<Phi>\<^sub>H (\<tau>\<^sub>0 a, F a)" "H (\<tau>\<^sub>0 a) \<star>\<^sub>D H (\<tau>.F.unit a)"
                           "H (\<tau>\<^sub>0 a \<star>\<^sub>C \<tau>.F.unit a)" "\<Phi>\<^sub>H (\<tau>\<^sub>0 a, \<tau>.F.map\<^sub>0 a)"]
               by auto
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 a) \<star>\<^sub>D H (\<tau>.F.unit a)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 a) \<star>\<^sub>D H.unit (\<tau>.F.map\<^sub>0 a)) \<cdot>\<^sub>D
                          \<r>\<^sub>D\<^sup>-\<^sup>1[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D
                          (H (\<l>\<^sub>C[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a)) \<cdot>\<^sub>D
                          (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
         proof -
           have "D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 a, \<tau>.F.map\<^sub>0 a)) \<cdot>\<^sub>D H (\<r>\<^sub>C\<^sup>-\<^sup>1[\<tau>\<^sub>0 a])
                   = (H (\<tau>\<^sub>0 a) \<star>\<^sub>D H.unit (\<tau>.F.map\<^sub>0 a)) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[H (\<tau>\<^sub>0 a)]"
             using a H.preserves_runit(2) [of "\<tau>\<^sub>0 a"] D.comp_assoc
             by (metis C.ideD(1) C.runit'_simps(1) C.src.preserves_ide C.trg_src
                 D.invert_side_of_triangle(1) \<tau>.F.map\<^sub>0_def H.cmp_components_are_iso
                 H.preserves_reflects_arr \<tau>.ide_map\<^sub>0_obj \<tau>.map\<^sub>0_simps(2))
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 a) \<star>\<^sub>D H (\<tau>.F.unit a)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 a) \<star>\<^sub>D H.unit (\<tau>.F.map\<^sub>0 a)) \<cdot>\<^sub>D
                          \<r>\<^sub>D\<^sup>-\<^sup>1[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D
                          \<l>\<^sub>D[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D
                          (D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<star>\<^sub>D H (\<tau>\<^sub>0 a)) \<cdot>\<^sub>D
                          (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
         proof -
           have "H (\<l>\<^sub>C[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a)
                   = \<l>\<^sub>D[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D (D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<star>\<^sub>D H (\<tau>\<^sub>0 a)) \<cdot>\<^sub>D
                     D.inv (\<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a)) \<cdot>\<^sub>D \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a)"
             using a H.preserves_lunit(1) D.comp_assoc by auto
           also have "... = \<l>\<^sub>D[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D (D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
           proof - 
             have "D.inv (\<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a)) \<cdot>\<^sub>D \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a)
                     = H (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a)"
               using a H.cmp_components_are_iso D.comp_inv_arr'
               by (metis C.isomorphic_implies_hpar(1) \<tau>.G.map\<^sub>0_simps(2) \<tau>.G.weakly_preserves_objects
                   H.cmp_simps(4) \<tau>.ide_map\<^sub>0_obj \<tau>.map\<^sub>0_simps(3))
             moreover have "(D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<star>\<^sub>D H (\<tau>\<^sub>0 a)) \<cdot>\<^sub>D
                            (H (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))
                              = (D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
               using a H.unit_char(2) D.comp_arr_dom
               by (metis D.arr_inv D.dom_inv D.whisker_right \<tau>.G.map\<^sub>0_simps(1) H.unit_simps(5)
                   H.preserves_ide \<tau>.ide_map\<^sub>0_obj)
             ultimately show ?thesis by simp
           qed
           finally have "H (\<l>\<^sub>C[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D \<Phi>\<^sub>H (\<tau>.G.map\<^sub>0 a, \<tau>\<^sub>0 a)
                           = \<l>\<^sub>D[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D (D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<star>\<^sub>D H (\<tau>\<^sub>0 a))"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = ((H (\<tau>\<^sub>0 a) \<star>\<^sub>D H (\<tau>.F.unit a)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 a) \<star>\<^sub>D H.unit (\<tau>.F.map\<^sub>0 a))) \<cdot>\<^sub>D
                          \<r>\<^sub>D\<^sup>-\<^sup>1[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D
                          \<l>\<^sub>D[H (\<tau>\<^sub>0 a)]"
         proof -
           have "(D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<star>\<^sub>D H (\<tau>\<^sub>0 a)) \<cdot>\<^sub>D
                 (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))
                   = D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<cdot>\<^sub>D H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a)"
             using a D.whisker_right H.unit_char(2) by simp
           also have "... = H.map\<^sub>0 (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a)"
             using a H.unit_char(1-2) [of "\<tau>.G.map\<^sub>0 a"] D.comp_inv_arr' by simp
           finally have "(D.inv (H.unit (\<tau>.G.map\<^sub>0 a)) \<star>\<^sub>D H (\<tau>\<^sub>0 a)) \<cdot>\<^sub>D
                         (H.unit (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a))
                           = H.map\<^sub>0 (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a)"
             by blast
           moreover have "\<l>\<^sub>D[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D (H.map\<^sub>0 (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a)) = \<l>\<^sub>D[H (\<tau>\<^sub>0 a)]"
             using a D.comp_arr_dom [of "\<l>\<^sub>D[H (\<tau>\<^sub>0 a)]" "H.map\<^sub>0 (\<tau>.G.map\<^sub>0 a) \<star>\<^sub>D H (\<tau>\<^sub>0 a)"]
             by auto
           ultimately show ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 a) \<star>\<^sub>D HoF.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[H (\<tau>\<^sub>0 a)] \<cdot>\<^sub>D \<l>\<^sub>D[H (\<tau>\<^sub>0 a)]"
           using a HoF.unit_char' D.whisker_left [of "H (\<tau>\<^sub>0 a)"] by simp
         also have "... = (map\<^sub>0 a \<star>\<^sub>D HoF.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a]"
           unfolding map\<^sub>0_def by simp
         finally show "(map\<^sub>0 a \<star>\<^sub>D HoF.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[map\<^sub>0 a] =
                       map\<^sub>1 a \<cdot>\<^sub>D (HoG.unit a \<star>\<^sub>D map\<^sub>0 a)"
           by simp
       qed
       show "\<And>f g. \<lbrakk>B.ide f; B.ide g; src\<^sub>B g = trg\<^sub>B f\<rbrakk> \<Longrightarrow>
                     (map\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>D HoF.cmp (g, f)) \<cdot>\<^sub>D
                     \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>B g), (H \<circ> F) g, (H \<circ> F) f] \<cdot>\<^sub>D
                     (map\<^sub>1 g \<star>\<^sub>D (H \<circ> F) f) \<cdot>\<^sub>D
                     D.inv \<a>\<^sub>D[(H \<circ> G) g, map\<^sub>0 (src\<^sub>B g), (H \<circ> F) f] \<cdot>\<^sub>D
                     ((H \<circ> G) g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[(H \<circ> G) g, (H \<circ> G) f, map\<^sub>0 (src\<^sub>B f)]
                       = map\<^sub>1 (g \<star>\<^sub>B f) \<cdot>\<^sub>D (HoG.cmp (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>B f))"
       proof -
         fix f g 
         assume f: "B.ide f" and g: "B.ide g" and fg: "src\<^sub>B g = trg\<^sub>B f"
         have "map\<^sub>1 (g \<star>\<^sub>B f) \<cdot>\<^sub>D (HoG.cmp (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>B f))
                 = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                   H (\<tau>\<^sub>1 (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                   \<Phi>\<^sub>H (G (g \<star>\<^sub>B f), \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                   (H (G (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                   H (\<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                   \<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
           unfolding map\<^sub>0_def map\<^sub>1_def HoG.cmp_def
           using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char D.comp_assoc by simp
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G (g \<star>\<^sub>B f), \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          ((H (G (g \<star>\<^sub>B f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          (H (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
           using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char
                 C.VV.arr_char C.VV.dom_char C.VV.cod_char \<tau>.G.FF_def
                 D.comp_assoc D.whisker_right
           by auto
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G (g \<star>\<^sub>B f), \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (H (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
         proof -
           have "(H (G (g \<star>\<^sub>B f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D (H (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))
                   = H (G (g \<star>\<^sub>B f)) \<cdot>\<^sub>D H (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))"
             using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char D.whisker_right
             by simp
           also have "... = H (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))"
             using f g fg D.comp_cod_arr [of "H (\<Phi>\<^sub>G (g, f))" "H (G (g \<star>\<^sub>B f))"]
                   B.VV.arr_char B.VV.dom_char B.VV.cod_char
             by simp
           finally have "(H (G (g \<star>\<^sub>B f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                         (H (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))
                           = H (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>1 (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                          H (\<Phi>\<^sub>G (g, f) \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
           using f g fg H.\<Phi>.naturality [of "(\<Phi>\<^sub>G (g, f), \<tau>\<^sub>0 (src\<^sub>B f))"]
                 B.VV.arr_char B.VV.dom_char B.VV.cod_char \<tau>.G.FF_def
                 C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
                 D.comp_assoc
           by simp
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 (g \<star>\<^sub>B f) \<cdot>\<^sub>C (\<Phi>\<^sub>G (g, f) \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
           using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char D.comp_assoc by simp
         also have "... = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          H ((\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>C
                             \<a>\<^sub>C[\<tau>\<^sub>0 (trg\<^sub>B g), F g, F f] \<cdot>\<^sub>C
                             (\<tau>\<^sub>1 g \<star>\<^sub>C F f) \<cdot>\<^sub>C
                             \<a>\<^sub>C\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 (src\<^sub>B g), F f] \<cdot>\<^sub>C
                             (G g \<star>\<^sub>C \<tau>\<^sub>1 f) \<cdot>\<^sub>C
                             \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)]) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
           using f g fg \<tau>.respects_hcomp by simp
         also have "... = (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C \<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          H \<a>\<^sub>C[\<tau>\<^sub>0 (trg\<^sub>B g), F g, F f] \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 g \<star>\<^sub>C F f) \<cdot>\<^sub>D
                          H (\<a>\<^sub>C\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 (src\<^sub>B g), F f]) \<cdot>\<^sub>D
                          H (G g \<star>\<^sub>C \<tau>\<^sub>1 f) \<cdot>\<^sub>D
                          H \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)] \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
           using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char \<tau>.F.FF_def D.comp_assoc
           by simp
         also have "... = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g \<star>\<^sub>C F f)) \<cdot>\<^sub>D
                          H \<a>\<^sub>C[\<tau>\<^sub>0 (trg\<^sub>B g), F g, F f]) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 g \<star>\<^sub>C F f) \<cdot>\<^sub>D
                          H (\<a>\<^sub>C\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 (src\<^sub>B g), F f]) \<cdot>\<^sub>D
                          H (G g \<star>\<^sub>C \<tau>\<^sub>1 f) \<cdot>\<^sub>D
                          H \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)] \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
         proof -
           have "D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<cdot>\<^sub>D H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C \<Phi>\<^sub>F (g, f))
                   = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g \<star>\<^sub>C F f))"
           proof -
             have "\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f)) \<cdot>\<^sub>D (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) =
                   H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g \<star>\<^sub>C F f)"
               using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char
                     C.VV.arr_char C.VV.dom_char C.VV.cod_char \<tau>.F.FF_def H.FF_def
                     H.\<Phi>.naturality [of "(\<tau>\<^sub>0 (trg\<^sub>B g), \<Phi>\<^sub>F (g, f))"]
               by auto
             moreover have "D.seq (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f)))"
               using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char
                     C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
               by auto
             moreover have "D.iso (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F (g \<star>\<^sub>B f))) \<and>
                            D.iso (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g \<star>\<^sub>C F f))"
               using f g fg H.cmp_components_are_iso by simp
             ultimately show ?thesis
             using f g fg D.invert_opposite_sides_of_square by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f)) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 g \<star>\<^sub>C F f)) \<cdot>\<^sub>D
                          H (\<a>\<^sub>C\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 (src\<^sub>B g), F f]) \<cdot>\<^sub>D
                          H (G g \<star>\<^sub>C \<tau>\<^sub>1 f) \<cdot>\<^sub>D
                          H \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)] \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
         proof -
           have "D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g \<star>\<^sub>C F f)) \<cdot>\<^sub>D H \<a>\<^sub>C[\<tau>\<^sub>0 (trg\<^sub>B g), F g, F f]
                   = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f)) \<cdot>\<^sub>D
                     \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                     (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                     D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f))"
           proof -
             have "D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g \<star>\<^sub>C F f)) \<cdot>\<^sub>D H \<a>\<^sub>C[\<tau>\<^sub>0 (trg\<^sub>B g), F g, F f]
                     = ((D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g \<star>\<^sub>C F f)) \<cdot>\<^sub>D
                       \<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g \<star>\<^sub>C F f)) \<cdot>\<^sub>D
                       (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f))) \<cdot>\<^sub>D
                       \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                       (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                       D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f))"
               using f g fg H.preserves_assoc(1) D.comp_assoc by simp
             also have "... = ((H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (F g \<star>\<^sub>C F f)) \<cdot>\<^sub>D
                              (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f))) \<cdot>\<^sub>D
                              \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                              (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                              D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f))"
               using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char
                     H.FF_def D.comp_assoc D.comp_inv_arr' H.cmp_components_are_iso
               by simp
             also have "... = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f)) \<cdot>\<^sub>D
                              \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                              (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                              D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f))"
               using f g fg C.VV.arr_char H.cmp_simps(5) D.comp_cod_arr D.comp_assoc by auto
             finally show ?thesis by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f)) \<cdot>\<^sub>D
                          H (\<a>\<^sub>C\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 (src\<^sub>B g), F f])) \<cdot>\<^sub>D
                          H (G g \<star>\<^sub>C \<tau>\<^sub>1 f) \<cdot>\<^sub>D
                          H \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)] \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
         proof -
           have "D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f)) \<cdot>\<^sub>D H (\<tau>\<^sub>1 g \<star>\<^sub>C F f)
                   = (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f))"
           proof -
             have "\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f) \<cdot>\<^sub>D (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f))
                     = H (\<tau>\<^sub>1 g \<star>\<^sub>C F f) \<cdot>\<^sub>D \<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f)"
               using f g fg H.\<Phi>.naturality [of "(\<tau>\<^sub>1 g, F f)"]
                     C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
               by simp
             moreover have "D.seq (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f))
                                  (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f))"
               using f g fg by fastforce
             moreover have "D.iso (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>C F g, F f)) \<and>
                            D.iso (\<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f))"
               using f g fg H.cmp_components_are_iso by simp
             ultimately show ?thesis
               using f g fg D.invert_opposite_sides_of_square by metis
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (G g), H (\<tau>\<^sub>0 (src\<^sub>B g)), H (F f)] \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f))) \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g) \<star>\<^sub>C F f)) \<cdot>\<^sub>D
                          H (G g \<star>\<^sub>C \<tau>\<^sub>1 f)) \<cdot>\<^sub>D
                          H \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)] \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
         proof -
           have "D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f)) \<cdot>\<^sub>D H (\<a>\<^sub>C\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 (src\<^sub>B g), F f])
                   = (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[H (G g), H (\<tau>\<^sub>0 (src\<^sub>B g)), H (F f)] \<cdot>\<^sub>D
                     (H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f))) \<cdot>\<^sub>D
                     D.inv (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g) \<star>\<^sub>C F f))"
           proof -
             have "D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f)) \<cdot>\<^sub>D H (\<a>\<^sub>C\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 (src\<^sub>B g), F f])
                     = ((D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f)) \<cdot>\<^sub>D
                       \<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f)) \<cdot>\<^sub>D
                       (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f))) \<cdot>\<^sub>D
                       \<a>\<^sub>D\<^sup>-\<^sup>1[H (G g), H (\<tau>\<^sub>0 (src\<^sub>B g)), H (F f)] \<cdot>\<^sub>D
                       (H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f))) \<cdot>\<^sub>D
                       D.inv (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g) \<star>\<^sub>C F f))"
               using f g fg H.preserves_assoc(2) D.comp_assoc by simp
             also have "... = (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[H (G g), H (\<tau>\<^sub>0 (src\<^sub>B g)), H (F f)] \<cdot>\<^sub>D
                              (H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f))) \<cdot>\<^sub>D
                              D.inv (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g) \<star>\<^sub>C F f))"
             proof -
               have "D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f)) \<cdot>\<^sub>D \<Phi>\<^sub>H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g), F f)
                       = H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)"
                 using f g fg H.cmp_components_are_iso D.comp_inv_arr' H.FF_def
                       C.VV.arr_char C.VV.dom_char C.VV.cod_char
                 by simp
               moreover have "(H (G g \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f))
                                = (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f))"
                 using f g fg D.comp_cod_arr C.VV.arr_char C.VV.dom_char C.VV.cod_char
                 by simp
               ultimately show ?thesis by simp
             qed
             finally show ?thesis by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (G g), H (\<tau>\<^sub>0 (src\<^sub>B g)), H (F f)] \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f))) \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D H (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          H \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)]) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
         proof -
           have "D.inv (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g) \<star>\<^sub>C F f)) \<cdot>\<^sub>D H (G g \<star>\<^sub>C \<tau>\<^sub>1 f)
                   = (H (G g) \<star>\<^sub>D H (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f)))"
           proof -
             have "\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (trg\<^sub>B f) \<star>\<^sub>C F f) \<cdot>\<^sub>D (H (G g) \<star>\<^sub>D H (\<tau>\<^sub>1 f))
                     = H (G g \<star>\<^sub>C \<tau>\<^sub>1 f) \<cdot>\<^sub>D \<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))"
               using f g fg H.\<Phi>.naturality [of "(G g, \<tau>\<^sub>1 f)"]
                     C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
               by simp
             thus ?thesis
               using f g fg H.cmp_components_are_iso
                     C.VV.arr_char C.VV.dom_char C.VV.cod_char
                     D.invert_opposite_sides_of_square
                       [of "\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (trg\<^sub>B f) \<star>\<^sub>C F f)" "H (G g) \<star>\<^sub>D H (\<tau>\<^sub>1 f)"
                           "H (G g \<star>\<^sub>C \<tau>\<^sub>1 f)" "\<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))"]
               by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (G g), H (\<tau>\<^sub>0 (src\<^sub>B g)), H (F f)] \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f))) \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D H (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (G g), H (G f), H (\<tau>\<^sub>0 (src\<^sub>B f))] \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (G g, G f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))"
         proof -
           have "D.inv (\<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D H \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)]
                   = (H (G g) \<star>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                     \<a>\<^sub>D[H (G g), H (G f), H (\<tau>\<^sub>0 (src\<^sub>B f))] \<cdot>\<^sub>D
                     (D.inv (\<Phi>\<^sub>H (G g, G f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                     D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)))"
           proof -
             have "D.inv (\<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D H \<a>\<^sub>C[G g, G f, \<tau>\<^sub>0 (src\<^sub>B f)]
                     = ((D.inv (\<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                       \<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                       (H (G g) \<star>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f)))) \<cdot>\<^sub>D
                       \<a>\<^sub>D[H (G g), H (G f), H (\<tau>\<^sub>0 (src\<^sub>B f))] \<cdot>\<^sub>D
                       (D.inv (\<Phi>\<^sub>H (G g, G f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                       D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f)))"
               using f g fg H.preserves_assoc(1)
                     H.cmp_components_are_iso C.VV.arr_char C.VV.dom_char C.VV.cod_char
                     D.comp_assoc
               by simp
             moreover have "D.inv (\<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                            \<Phi>\<^sub>H (G g, G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))
                              = H (G g) \<star>\<^sub>D H (G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))"
               using f g fg H.cmp_components_are_iso C.VV.arr_char C.VV.dom_char C.VV.cod_char
                     D.comp_inv_arr' H.FF_def
               by simp
             moreover have "(H (G g) \<star>\<^sub>D H (G f \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                            (H (G g) \<star>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f)))
                              = H (G g) \<star>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f))"
               using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char
                     D.whisker_left D.comp_cod_arr
               by simp
             ultimately show ?thesis by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = ((H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f))) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                          ((D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f))) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (G g), H (\<tau>\<^sub>0 (src\<^sub>B g)), H (F f)] \<cdot>\<^sub>D
                          ((H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f))) \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D H (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f)))) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (G g), H (G f), H (\<tau>\<^sub>0 (src\<^sub>B f))]"
         proof - 
           have "\<a>\<^sub>D[H (G g), H (G f), H (\<tau>\<^sub>0 (src\<^sub>B f))] \<cdot>\<^sub>D
                 (D.inv (\<Phi>\<^sub>H (G g, G f)) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                 (D.inv (\<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                 \<Phi>\<^sub>H (G g \<star>\<^sub>C G f, \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                 (\<Phi>\<^sub>H (G g, G f) \<star>\<^sub>D H (\<tau>\<^sub>0 (src\<^sub>B f)))
                   = \<a>\<^sub>D[H (G g), H (G f), H (\<tau>\<^sub>0 (src\<^sub>B f))]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.whisker_right [of "H (\<tau>\<^sub>0 (src\<^sub>B f))" "D.inv (\<Phi>\<^sub>H (G g, G f))" "\<Phi>\<^sub>H (G g, G f)"]
                   D.comp_inv_arr' C.VV.arr_char C.VV.dom_char C.VV.cod_char H.FF_def
                   H.cmp_components_are_iso
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (F (g \<star>\<^sub>B f)) \<cdot>\<^sub>D
                          H (\<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (F g, F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (\<tau>\<^sub>0 (trg\<^sub>B g)), H (F g), H (F f)] \<cdot>\<^sub>D
                          (D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 g) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[H (G g), H (\<tau>\<^sub>0 (src\<^sub>B g)), H (F f)] \<cdot>\<^sub>D
                          (H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f)) \<cdot>\<^sub>D
                          H (\<tau>\<^sub>1 f) \<cdot>\<^sub>D
                          \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f))) \<cdot>\<^sub>D
                          \<a>\<^sub>D[H (G g), H (G f), H (\<tau>\<^sub>0 (src\<^sub>B f))]"
         proof -
           have "(H (G g) \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f))) \<cdot>\<^sub>D
                 (H (G g) \<star>\<^sub>D H (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D
                 (H (G g) \<star>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f)))
                   = H (G g) \<star>\<^sub>D
                       D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (src\<^sub>B g), F f)) \<cdot>\<^sub>D H (\<tau>\<^sub>1 f) \<cdot>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f))"
             using f g fg C.VV.arr_char C.VV.dom_char C.VV.cod_char D.whisker_left
                   H.cmp_components_are_iso
             by simp
           moreover have "(H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f))
                            = H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D
                                H (F (g \<star>\<^sub>B f)) \<cdot>\<^sub>D H (\<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<Phi>\<^sub>H (F g, F f)"
           proof -
             have "(H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D (H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D \<Phi>\<^sub>H (F g, F f))
                     = H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D H (\<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<Phi>\<^sub>H (F g, F f)"
             proof -
               have "D.arr (H (\<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<Phi>\<^sub>H (F g, F f))"
                 using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char \<tau>.F.FF_def
                 by (intro D.seqI) auto
               thus ?thesis
                 using f g fg D.whisker_left C.VV.arr_char by simp
             qed
             also have "... = H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D
                                (H (F (g \<star>\<^sub>B f) \<cdot>\<^sub>C \<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D \<Phi>\<^sub>H (F g, F f)"
               using f g fg C.comp_cod_arr B.VV.arr_char \<tau>.F.cmp_simps(5) by auto
             also have "... = H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D
                                (H (F (g \<star>\<^sub>B f)) \<cdot>\<^sub>D H (\<Phi>\<^sub>F (g, f))) \<cdot>\<^sub>D \<Phi>\<^sub>H (F g, F f)"
               using f g fg B.VV.arr_char \<tau>.F.cmp_simps(5) by auto
             also have "... = H (\<tau>\<^sub>0 (trg\<^sub>B g)) \<star>\<^sub>D
                                H (F (g \<star>\<^sub>B f)) \<cdot>\<^sub>D H (\<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<Phi>\<^sub>H (F g, F f)"
               using D.comp_assoc by simp
             finally show ?thesis by simp
           qed
           moreover have "(D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (H (\<tau>\<^sub>1 g) \<star>\<^sub>D H (F f)) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f))
                            = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B g), F g)) \<cdot>\<^sub>D
                              H (\<tau>\<^sub>1 g) \<cdot>\<^sub>D \<Phi>\<^sub>H (G g, \<tau>\<^sub>0 (src\<^sub>B g)) \<star>\<^sub>D H (F f)"
             using f g fg D.whisker_right C.VV.arr_char C.VV.dom_char C.VV.cod_char
             by simp
           ultimately show ?thesis
             by simp
         qed
         also have "... = (map\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>D HoF.cmp (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>B g), (H \<circ> F) g, (H \<circ> F) f] \<cdot>\<^sub>D
                          (map\<^sub>1 g \<star>\<^sub>D (H \<circ> F) f) \<cdot>\<^sub>D
                          D.inv \<a>\<^sub>D[(H \<circ> G) g, map\<^sub>0 (src\<^sub>B g), (H \<circ> F) f] \<cdot>\<^sub>D
                          ((H \<circ> G) g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[(H \<circ> G) g, (H \<circ> G) f, map\<^sub>0 (src\<^sub>B f)]"
           unfolding map\<^sub>0_def map\<^sub>1_def HoF.cmp_def
           using f g fg B.VV.arr_char B.VV.dom_char B.VV.cod_char by simp
         finally show "(map\<^sub>0 (trg\<^sub>B g) \<star>\<^sub>D HoF.cmp (g, f)) \<cdot>\<^sub>D
                       \<a>\<^sub>D[map\<^sub>0 (trg\<^sub>B g), (H \<circ> F) g, (H \<circ> F) f] \<cdot>\<^sub>D
                       (map\<^sub>1 g \<star>\<^sub>D (H \<circ> F) f) \<cdot>\<^sub>D
                       D.inv \<a>\<^sub>D[(H \<circ> G) g, map\<^sub>0 (src\<^sub>B g), (H \<circ> F) f] \<cdot>\<^sub>D
                       ((H \<circ> G) g \<star>\<^sub>D map\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[(H \<circ> G) g, (H \<circ> G) f, map\<^sub>0 (src\<^sub>B f)]
                         = map\<^sub>1 (g \<star>\<^sub>B f) \<cdot>\<^sub>D (HoG.cmp (g, f) \<star>\<^sub>D map\<^sub>0 (src\<^sub>B f))"
           by simp
       qed
     qed
 
     lemma is_pseudonatural_transformation:
     shows "pseudonatural_transformation V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
              HoF.map HoF.cmp HoG.map HoG.cmp map\<^sub>0 map\<^sub>1"
       ..
 
   end
 
   subsection "Pseudonatural Equivalences"
 
   text \<open>
     A \emph{pseudonatural equivalence} is a pseudonatural transformation whose components
     at objects are equivalence maps.  Pseudonatural equivalences between pseudofunctors
     generalize natural isomorphisms between ordinary functors.
   \<close>
 
   locale pseudonatural_equivalence =
     pseudonatural_transformation +
   assumes components_are_equivalences: "C.obj a \<Longrightarrow> D.equivalence_map (\<tau>\<^sub>0 a)"
 
   subsubsection "Identity Transformations are Pseudonatural Equivalences"
 
   sublocale identity_pseudonatural_transformation \<subseteq>
             pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
               F \<Phi>\<^sub>F F \<Phi>\<^sub>F map\<^sub>0 map\<^sub>1
     by unfold_locales (simp add: D.obj_is_equivalence_map)
 
   subsubsection "Composition of Pseudonatural Equivalences"
 
   locale composite_pseudonatural_equivalence =
     composite_pseudonatural_transformation +
     \<sigma>: pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
          F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<sigma>\<^sub>0 \<sigma>\<^sub>1 +
     \<rho>: pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
          G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<rho>\<^sub>0 \<rho>\<^sub>1
   begin
 
     sublocale pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                 F \<Phi>\<^sub>F H \<Phi>\<^sub>H map\<^sub>0 map\<^sub>1
       apply unfold_locales
       by (metis D.equivalence_maps_compose D.hcomp_in_hhomE \<rho>.components_are_equivalences
           \<sigma>.components_are_equivalences map\<^sub>0_def map\<^sub>0_in_hom(1))
 
     lemma is_pseudonatural_equivalence:
     shows "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
              F \<Phi>\<^sub>F H \<Phi>\<^sub>H map\<^sub>0 map\<^sub>1"
       ..
 
   end
 
   locale pseudonatural_equivalence_whisker_right =
     pseudonatural_transformation_whisker_right +
     \<tau>: pseudonatural_equivalence
          V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
   begin
 
     interpretation FoH: composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                           V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H F \<Phi>\<^sub>F
       ..
     interpretation GoH: composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                           V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H G \<Phi>\<^sub>G
       ..
 
     sublocale pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                           \<open>F o H\<close> FoH.cmp \<open>G o H\<close> GoH.cmp map\<^sub>0 map\<^sub>1
       using map\<^sub>0_def \<tau>.components_are_equivalences
       by unfold_locales simp
 
     lemma is_pseudonatural_equivalence:
     shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                           (F o H) FoH.cmp (G o H) GoH.cmp map\<^sub>0 map\<^sub>1"
       ..
 
   end
 
   locale pseudonatural_equivalence_whisker_left =
     pseudonatural_transformation_whisker_left +
     \<tau>: pseudonatural_equivalence
          V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
   begin
 
     interpretation HoF: composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                           V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F H \<Phi>\<^sub>H
       ..
     interpretation HoG: composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                           V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G H \<Phi>\<^sub>H
       ..
 
     sublocale pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                           \<open>H o F\<close> HoF.cmp \<open>H o G\<close> HoG.cmp map\<^sub>0 map\<^sub>1
       using map\<^sub>0_def \<tau>.components_are_equivalences H.preserves_equivalence_maps
       by unfold_locales simp
 
     lemma is_pseudonatural_equivalence:
     shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                           (H o F) HoF.cmp (H o G) HoG.cmp map\<^sub>0 map\<^sub>1"
       ..
 
   end
 
   subsubsection "Converse of a Pseudonatural Equivalence"
 
   text \<open>
     It is easy to see that natural isomorphism between ordinary functors is a symmetric
     relation because a unique inverse to a natural isomorphism is obtained merely by inverting
     the components.  However the situation is more difficult for pseudonatural equivalences
     because they do not have unique inverses.  Instead, we have to choose a quasi-inverse for
     each of the components.  In order to satisfy the required coherence conditions,
     it is necessary for these quasi-inverses to be part of chosen adjoint equivalences.
     Some long calculations to establish the coherence conditions seem unavoidable.
     The purpose of this section is to carry out the construction, given a pseudonatural
     equivalence, of a ``converse'' pseudonatural equivalence in the opposite direction.
   \<close>
 
   locale converse_pseudonatural_equivalence =
     C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C +
     D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D +
     F: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F +
     G: pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G +
     \<tau>: pseudonatural_equivalence
   begin
 
     abbreviation (input) F\<^sub>0
     where "F\<^sub>0 \<equiv> F.map\<^sub>0"
 
     abbreviation (input) G\<^sub>0
     where "G\<^sub>0 \<equiv> G.map\<^sub>0"
 
     definition map\<^sub>0
     where "map\<^sub>0 a = (SOME g. \<exists>\<eta> \<epsilon>. adjoint_equivalence_in_bicategory
                                      V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) g \<eta> \<epsilon>)"
 
     abbreviation (input) \<tau>\<^sub>0'
     where "\<tau>\<^sub>0' \<equiv> map\<^sub>0"
 
     definition unit
     where "unit a = (SOME \<eta>. \<exists>\<epsilon>. adjoint_equivalence_in_bicategory
                                    V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (\<tau>\<^sub>0' a) \<eta> \<epsilon>)"
 
     abbreviation (input) \<eta>
     where "\<eta> \<equiv> unit"
 
     definition counit
     where "counit a = (SOME \<epsilon>. adjoint_equivalence_in_bicategory
                                  V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (\<tau>\<^sub>0' a) (\<eta> a) \<epsilon>)"
 
     abbreviation (input) \<epsilon>
     where "\<epsilon> \<equiv> counit"
 
     lemma chosen_adjoint_equivalence:
     assumes "C.obj a"
     shows "adjoint_equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (\<tau>\<^sub>0' a) (\<eta> a) (\<epsilon> a)"
       using assms \<tau>.components_are_equivalences map\<^sub>0_def unit_def counit_def
             D.obj_is_equivalence_map D.equivalence_map_extends_to_adjoint_equivalence
             someI_ex [of "\<lambda>g. \<exists>\<eta> \<epsilon>. adjoint_equivalence_in_bicategory
                                       V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) g \<eta> \<epsilon>"]
             someI_ex [of "\<lambda>\<eta>. \<exists>\<epsilon>. adjoint_equivalence_in_bicategory
                                     V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (\<tau>\<^sub>0' a) \<eta> \<epsilon>"]
             someI_ex [of "\<lambda>\<epsilon>. adjoint_equivalence_in_bicategory
                                     V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (\<tau>\<^sub>0' a) (\<eta> a) \<epsilon>"]
       by simp
 
     lemma map\<^sub>0_in_hhom [intro]:
     assumes "C.obj a"
     shows "\<guillemotleft>\<tau>\<^sub>0' a : G\<^sub>0 a \<rightarrow>\<^sub>D F\<^sub>0 a\<guillemotright>"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show ?thesis
         using assms \<tau>.map\<^sub>0_in_hhom [of a] antipar by auto
     qed
 
     lemma map\<^sub>0_simps [simp]:
     assumes "C.obj a"
     shows "D.ide (\<tau>\<^sub>0' a)" and "src\<^sub>D (\<tau>\<^sub>0' a) = G\<^sub>0 a" and "trg\<^sub>D (\<tau>\<^sub>0' a) = F\<^sub>0 a"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show "D.ide (\<tau>\<^sub>0' a)"
         by simp
       show "src\<^sub>D (\<tau>\<^sub>0' a) = G\<^sub>0 a"
         using assms map\<^sub>0_in_hhom by blast
       show "trg\<^sub>D (\<tau>\<^sub>0' a) = F\<^sub>0 a"
         using assms map\<^sub>0_in_hhom by blast
     qed
 
     lemma equivalence_map_map\<^sub>0 [simp]:
     assumes "C.obj a"
     shows "D.equivalence_map (\<tau>\<^sub>0' a)"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show "D.equivalence_map (\<tau>\<^sub>0' a)"
         using D.equivalence_map_def adjoint_equivalence_in_bicategory_axioms
               dual_equivalence
         by blast
     qed
 
     lemma unit_in_hom [intro]:
     assumes "C.obj a"
     shows "\<guillemotleft>\<eta> a : F\<^sub>0 a \<rightarrow>\<^sub>D F\<^sub>0 a\<guillemotright>"
     and "\<guillemotleft>\<eta> a : F\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a\<guillemotright>"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show "\<guillemotleft>\<eta> a : F\<^sub>0 a \<rightarrow>\<^sub>D F\<^sub>0 a\<guillemotright>"
         using assms unit_in_hom antipar(1) by auto
       show "\<guillemotleft>\<eta> a : F\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a\<guillemotright>"
         using assms unit_in_hom antipar(1) by auto
     qed
 
     lemma unit_simps [simp]:
     assumes "C.obj a"
     shows "D.iso (\<eta> a)" and "D.arr (\<eta> a)"
     and "src\<^sub>D (\<eta> a) = F\<^sub>0 a" and "trg\<^sub>D (\<eta> a) = F\<^sub>0 a"
     and "D.dom (\<eta> a) = F\<^sub>0 a" and "D.cod (\<eta> a) = \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show "D.iso (\<eta> a)" and "D.arr (\<eta> a)"
       and "src\<^sub>D (\<eta> a) = F\<^sub>0 a" and "trg\<^sub>D (\<eta> a) = F\<^sub>0 a"
       and "D.dom (\<eta> a) = F\<^sub>0 a" and "D.cod (\<eta> a) = \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a"
         using assms by auto
     qed
 
     lemma iso_unit:
     assumes "C.obj a"
     shows "D.iso (\<eta> a)"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show ?thesis by simp
     qed
 
     lemma counit_in_hom [intro]:
     assumes "C.obj a"
     shows "\<guillemotleft>\<epsilon> a : G\<^sub>0 a \<rightarrow>\<^sub>D G\<^sub>0 a\<guillemotright>"
     and "\<guillemotleft>\<epsilon> a : \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a \<Rightarrow>\<^sub>D G\<^sub>0 a\<guillemotright>"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show "\<guillemotleft>\<epsilon> a : G\<^sub>0 a \<rightarrow>\<^sub>D G\<^sub>0 a\<guillemotright>"
         using assms counit_in_hom antipar(2) by auto
       show " \<guillemotleft>\<epsilon> a : \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a \<Rightarrow>\<^sub>D G\<^sub>0 a\<guillemotright>"
         using assms counit_in_hom antipar(2) by simp
     qed
 
     lemma counit_simps [simp]:
     assumes "C.obj a"
     shows "D.iso (\<epsilon> a)" and "D.arr (\<epsilon> a)"
     and "src\<^sub>D (\<epsilon> a) = G\<^sub>0 a" and "trg\<^sub>D (\<epsilon> a) = G\<^sub>0 a"
     and "D.dom (\<epsilon> a) = \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a" and "D.cod (\<epsilon> a) = G\<^sub>0 a"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show "D.iso (\<epsilon> a)" and "D.arr (\<epsilon> a)"
       and "src\<^sub>D (\<epsilon> a) = G\<^sub>0 a" and "trg\<^sub>D (\<epsilon> a) = G\<^sub>0 a"
       and "D.dom (\<epsilon> a) = \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a" and "D.cod (\<epsilon> a) = G\<^sub>0 a"
         using assms by auto
     qed
 
     lemma iso_counit:
     assumes "C.obj a"
     shows "D.iso (\<epsilon> a)"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show ?thesis by simp
     qed
 
     lemma quasi_inverts_components:
     assumes "C.obj a"
     shows "D.isomorphic (\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) (F\<^sub>0 a)"
     and "D.isomorphic (\<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) (G\<^sub>0 a)"
     and "D.quasi_inverses (\<tau>\<^sub>0 a) (\<tau>\<^sub>0' a)"
     proof -
       interpret adjoint_equivalence_in_bicategory
                   V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
         using assms chosen_adjoint_equivalence by simp
       show "D.isomorphic (\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) (F\<^sub>0 a)"
         using assms D.isomorphic_def D.isomorphic_symmetric unit_is_iso by blast
       show "D.isomorphic (\<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) (G\<^sub>0 a)"
         using assms D.isomorphic_def counit_is_iso by blast
       show "D.quasi_inverses (\<tau>\<^sub>0 a) (\<tau>\<^sub>0' a)"
         using D.quasi_inverses_def equivalence_in_bicategory_axioms by auto
     qed
 
     definition map\<^sub>1
     where "map\<^sub>1 f = (\<tau>\<^sub>0' (trg\<^sub>C f) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' (trg\<^sub>C f) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> (src\<^sub>C f)) \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' (trg\<^sub>C f) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 (src\<^sub>C f), \<tau>\<^sub>0' (src\<^sub>C f)]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' (trg\<^sub>C f), G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f), \<tau>\<^sub>0' (src\<^sub>C f)] \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' (trg\<^sub>C f) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f)) \<cdot>\<^sub>D
                     (\<a>\<^sub>D[\<tau>\<^sub>0' (trg\<^sub>C f), \<tau>\<^sub>0 (trg\<^sub>C f), F f] \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f)) \<cdot>\<^sub>D
                     ((\<eta> (trg\<^sub>C f) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f)) \<cdot>\<^sub>D
                     (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f))"
 
     abbreviation (input) \<tau>\<^sub>1'
     where "\<tau>\<^sub>1' \<equiv> map\<^sub>1"
 
     lemma map\<^sub>1_in_hom [intro]:
     assumes "C.ide f"
     shows "\<guillemotleft>\<tau>\<^sub>1' f : G\<^sub>0 (src\<^sub>C f) \<rightarrow>\<^sub>D F\<^sub>0 (trg\<^sub>C f)\<guillemotright>"
     and "\<guillemotleft>\<tau>\<^sub>1' f : F f \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f) \<Rightarrow>\<^sub>D \<tau>\<^sub>0' (trg\<^sub>C f) \<star>\<^sub>D G f\<guillemotright>"
     proof -
       show "\<guillemotleft>\<tau>\<^sub>1' f : F f \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f) \<Rightarrow>\<^sub>D \<tau>\<^sub>0' (trg\<^sub>C f) \<star>\<^sub>D G f\<guillemotright>"
         using assms \<tau>.iso_map\<^sub>1_ide \<tau>.map\<^sub>1_in_vhom
         by (unfold map\<^sub>1_def, intro D.comp_in_homI) auto
       thus "\<guillemotleft>\<tau>\<^sub>1' f : G\<^sub>0 (src\<^sub>C f) \<rightarrow>\<^sub>D F\<^sub>0 (trg\<^sub>C f)\<guillemotright>"
         using assms D.vconn_implies_hpar(1-2) by auto
     qed
 
     lemma map\<^sub>1_simps [simp]:
     assumes "C.ide f"
     shows "D.arr (\<tau>\<^sub>1' f)"
     and "src\<^sub>D (\<tau>\<^sub>1' f) = G\<^sub>0 (src\<^sub>C f)" and "trg\<^sub>D (\<tau>\<^sub>1' f) = F\<^sub>0 (trg\<^sub>C f)"
     and "D.dom (\<tau>\<^sub>1' f) = F f \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f)" and "D.cod (\<tau>\<^sub>1' f) = \<tau>\<^sub>0' (trg\<^sub>C f) \<star>\<^sub>D G f"
       using assms map\<^sub>1_in_hom by auto
     
     lemma iso_map\<^sub>1_ide:
     assumes "C.ide f"
     shows "D.iso (\<tau>\<^sub>1' f)"
     proof -
       interpret src: adjoint_equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                         \<open>\<tau>\<^sub>0 (src\<^sub>C f)\<close> \<open>\<tau>\<^sub>0' (src\<^sub>C f)\<close> \<open>\<eta> (src\<^sub>C f)\<close> \<open>\<epsilon> (src\<^sub>C f)\<close>
         using assms chosen_adjoint_equivalence by simp
       interpret trg: adjoint_equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                         \<open>\<tau>\<^sub>0 (trg\<^sub>C f)\<close> \<open>\<tau>\<^sub>0' (trg\<^sub>C f)\<close> \<open>\<eta> (trg\<^sub>C f)\<close> \<open>\<epsilon> (trg\<^sub>C f)\<close>
 
         using assms chosen_adjoint_equivalence by simp
       show ?thesis
         unfolding map\<^sub>1_def
         using assms \<tau>.iso_map\<^sub>1_ide
         by (intro D.isos_compose) auto
     qed
 
     interpretation EV: self_evaluation_map V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D ..
     notation EV.eval ("\<lbrace>_\<rbrace>")
 
     sublocale pseudonatural_equivalence
                      V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G F \<Phi>\<^sub>F \<tau>\<^sub>0' \<tau>\<^sub>1'
     proof
       show "\<And>a. C.obj a \<Longrightarrow> D.ide (\<tau>\<^sub>0' a)"
         using map\<^sub>0_simps(1) by simp
       show "\<And>f. C.ide f \<Longrightarrow> D.iso (\<tau>\<^sub>1' f)"
         using iso_map\<^sub>1_ide by simp
       show "\<And>a. C.obj a \<Longrightarrow> \<guillemotleft>\<tau>\<^sub>0' a : src\<^sub>D (G a) \<rightarrow>\<^sub>D src\<^sub>D (F a)\<guillemotright>"
         by fastforce
       show "\<And>f. C.ide f \<Longrightarrow> \<guillemotleft>\<tau>\<^sub>1' f : F f \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f) \<Rightarrow>\<^sub>D \<tau>\<^sub>0' (trg\<^sub>C f) \<star>\<^sub>D G f\<guillemotright>"
         by auto
       show "\<And>a. C.obj a \<Longrightarrow> D.equivalence_map (\<tau>\<^sub>0' a)"
         by simp
       show "\<And>\<mu>. C.arr \<mu> \<Longrightarrow> \<tau>\<^sub>1' (C.cod \<mu>) \<cdot>\<^sub>D (F \<mu> \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C \<mu>))
                                = (\<tau>\<^sub>0' (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu>) \<cdot>\<^sub>D \<tau>\<^sub>1' (C.dom \<mu>)"
       proof -
         fix \<mu>
         assume \<mu>: "C.arr \<mu>"
         let ?a = "src\<^sub>C \<mu>"
         let ?b = "trg\<^sub>C \<mu>"
         let ?f = "C.dom \<mu>"
         let ?g = "C.cod \<mu>"
         have "\<tau>\<^sub>1' (C.cod \<mu>) \<cdot>\<^sub>D (F \<mu> \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                        = (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?g \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?g)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?g] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?b \<star>\<^sub>D F ?g) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?g] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F \<mu> \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           unfolding map\<^sub>1_def
           using \<mu> D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?g \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?g)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?g] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<eta> ?b \<star>\<^sub>D F ?g) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((trg\<^sub>D (F \<mu>) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           using \<mu> D.whisker_right D.lunit'_naturality [of "F \<mu>"] D.comp_assoc
                 D.whisker_right [of "\<tau>\<^sub>0' ?a" "trg\<^sub>D (F \<mu>) \<star>\<^sub>D F \<mu>" "\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f]"]
           by simp
         also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?g \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?g)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?g] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "((\<eta> ?b \<star>\<^sub>D F ?g) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D ((trg\<^sub>D (F \<mu>) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           proof -
             have "((\<eta> ?b \<star>\<^sub>D F ?g) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D ((trg\<^sub>D (F \<mu>) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                     = (\<eta> ?b \<star>\<^sub>D F ?g) \<cdot>\<^sub>D (trg\<^sub>D (F \<mu>) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
               using \<mu> D.whisker_right D.obj_simps by simp
             also have "... = (\<eta> ?b \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
               using \<mu> D.interchange [of "\<eta> ?b" "trg\<^sub>D (F \<mu>)" "F ?g" "F \<mu>"]
                     D.comp_arr_dom D.comp_cod_arr
               by simp
             also have "... = ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D (\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
               using \<mu> D.interchange [of "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b" "\<eta> ?b" "F \<mu>" "F ?f"]
                     D.comp_arr_dom D.comp_cod_arr
               by simp
             also have "... = (((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
               using \<mu> D.whisker_right by simp
             finally show ?thesis by blast
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?g \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?g)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?g] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?g] \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using \<mu> D.whisker_right by simp
           also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using \<mu> D.assoc_naturality [of "\<tau>\<^sub>0' ?b" "\<tau>\<^sub>0 ?b" "F \<mu>"] by simp
           also have "... = ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using \<mu> D.whisker_right by simp
           finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?g] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?g \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?g)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?g)) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using \<mu> \<tau>.iso_map\<^sub>1_ide D.whisker_right by simp
           also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?g) \<cdot>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using \<mu> \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
           also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using \<mu> \<tau>.iso_map\<^sub>1_ide \<tau>.naturality
                   D.invert_opposite_sides_of_square
                     [of "\<tau>\<^sub>1 ?g" "G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a" "\<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>" "\<tau>\<^sub>1 ?f"]
             by simp
           also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using \<mu> \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
           also have "... = ((\<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using \<mu> \<tau>.iso_map\<^sub>1_ide D.whisker_right by simp
           finally have "((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?g)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F \<mu>) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = ((\<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           using \<mu> D.assoc_naturality [of "\<tau>\<^sub>0' ?b" "G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a" "\<tau>\<^sub>0' ?a"] D.comp_assoc
           by simp
         also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D ((G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using \<mu> D.whisker_left by simp
           also have "... = \<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D \<a>\<^sub>D[G ?f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
             using \<mu> D.assoc_naturality [of "G \<mu>" "\<tau>\<^sub>0 ?a" "\<tau>\<^sub>0' ?a"] by simp
           also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
             using \<mu> D.whisker_left by simp
           finally have "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?g, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D G\<^sub>0 ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<tau>\<^sub>0' ?b \<star>\<^sub>D (G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using \<mu> D.whisker_left by simp
           also have "... = \<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<epsilon> ?a"
             using \<mu> D.interchange [of "G ?g" "G \<mu>" "\<epsilon> ?a" "\<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                   D.comp_arr_dom D.comp_cod_arr
             by simp
           also have "... = \<tau>\<^sub>0' ?b \<star>\<^sub>D (G \<mu> \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D (G ?f \<star>\<^sub>D \<epsilon> ?a)"
             using \<mu> D.interchange [of "G \<mu>" "G ?f" "G\<^sub>0 ?a" "\<epsilon> ?a"]
                   D.comp_arr_dom D.comp_cod_arr
             by simp
           also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?f \<star>\<^sub>D \<epsilon> ?a)"
             using \<mu> D.whisker_left D.obj_simps by auto
           finally have "(\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?g \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?f \<star>\<^sub>D \<epsilon> ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu>) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?f]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D G ?f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G ?f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G ?f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?b \<star>\<^sub>D F ?f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D G\<^sub>0 ?a)
                   = \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g] \<cdot>\<^sub>D (G \<mu> \<star>\<^sub>D G\<^sub>0 ?a)"
             using \<mu> D.whisker_left D.obj_simps by simp
           also have "... = \<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<cdot>\<^sub>D \<r>\<^sub>D[G ?f]"
             using \<mu> D.runit_naturality [of "G \<mu>"] by simp
           also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu>) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?f])"
             using \<mu> D.whisker_left by simp
           finally have "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?g]) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu> \<star>\<^sub>D G\<^sub>0 ?a)
                           = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu>) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G ?f])"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu>) \<cdot>\<^sub>D \<tau>\<^sub>1' ?f"
           unfolding map\<^sub>1_def
           using \<mu> D.comp_assoc by simp
         finally show "\<tau>\<^sub>1' ?g \<cdot>\<^sub>D (F \<mu> \<star>\<^sub>D \<tau>\<^sub>0' ?a) = (\<tau>\<^sub>0' ?b \<star>\<^sub>D G \<mu>) \<cdot>\<^sub>D \<tau>\<^sub>1' ?f"
           by blast
       qed
       show "\<And>a. C.obj a \<Longrightarrow>
                   (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0' a] = \<tau>\<^sub>1' a \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D \<tau>\<^sub>0' a)"
       proof -
         fix a
         assume "C.obj a"
         hence a: "C.obj a \<and> C.arr a \<and> C.ide a \<and> src\<^sub>C a = a \<and> trg\<^sub>C a = a"
           by auto
         have 1: "D.ide (F\<^sub>0 a)"
           using a F.map\<^sub>0_simps(1) by blast
         have 2: "D.ide (G\<^sub>0 a)"
           using a G.map\<^sub>0_simps(1) by blast
         have "\<tau>\<^sub>1' a \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D \<tau>\<^sub>0' a)
                        = (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, G a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D D.inv (\<tau>\<^sub>1 a)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (F.unit a \<star>\<^sub>D \<tau>\<^sub>0' a)"
           unfolding map\<^sub>1_def
           using a D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, G a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D D.inv (\<tau>\<^sub>1 a)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (((\<eta> a \<star>\<^sub>D F a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((F\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D \<tau>\<^sub>0' a) = \<l>\<^sub>D\<^sup>-\<^sup>1[F a] \<cdot>\<^sub>D F.unit a \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a D.whisker_right by simp
           also have "... = (F\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a D.lunit'_naturality [of "F.unit a"] by simp
           also have "... = ((F\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
             using a 1 D.whisker_right by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D \<tau>\<^sub>0' a) =
                         ((F\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, G a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D D.inv (\<tau>\<^sub>1 a)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "((\<eta> a \<star>\<^sub>D F a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D ((F\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                   = (\<eta> a \<star>\<^sub>D F a) \<cdot>\<^sub>D (F\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a 1 D.whisker_right by simp
           also have "... = (\<eta> a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a D.interchange [of "\<eta> a" "F\<^sub>0 a" "F a" "F.unit a"]
                   D.comp_cod_arr D.comp_arr_dom
             by simp
           also have "... = ((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D (\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a D.interchange [of "\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a" "\<eta> a" "F.unit a" "F\<^sub>0 a"]
                   D.comp_cod_arr D.comp_arr_dom
             by simp
           also have "... = (((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)"
             using a 1 D.whisker_right by simp
           finally have "((\<eta> a \<star>\<^sub>D F a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D ((F\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                           = (((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, G a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' a \<star>\<^sub>D D.inv (\<tau>\<^sub>1 a)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "(\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                   = \<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F a] \<cdot>\<^sub>D ((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a D.whisker_right by simp
           also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a D.assoc_naturality [of "\<tau>\<^sub>0' a" "\<tau>\<^sub>0 a" "F.unit a"] by simp
           also have "... = ((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                              (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
             using a 1 D.whisker_right by simp
           finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                           = ((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, G a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "((\<tau>\<^sub>0' a \<star>\<^sub>D D.inv (\<tau>\<^sub>1 a)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D ((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                   = (\<tau>\<^sub>0' a \<star>\<^sub>D D.inv (\<tau>\<^sub>1 a)) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a \<tau>.iso_map\<^sub>1_ide D.whisker_right by simp
           also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D D.inv (\<tau>\<^sub>1 a) \<cdot>\<^sub>D (\<tau>\<^sub>0 a \<star>\<^sub>D F.unit a)) \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
           also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a"
           proof -
             have "(\<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0 a] = \<tau>\<^sub>1 a \<cdot>\<^sub>D (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a)"
               using a \<tau>.respects_unit by simp
             hence "D.inv (\<tau>\<^sub>1 a) \<cdot>\<^sub>D (\<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0 a]
                      = G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a"
               using a \<tau>.iso_map\<^sub>1_ide D.invert_side_of_triangle(1) by simp
             hence "D.inv (\<tau>\<^sub>1 a) \<cdot>\<^sub>D (\<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]
                      = (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]"
               using a D.iso_lunit D.invert_side_of_triangle(2) D.comp_assoc by simp
             hence "D.inv (\<tau>\<^sub>1 a) \<cdot>\<^sub>D (\<tau>\<^sub>0 a \<star>\<^sub>D F.unit a)
                      = (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]"
               using a D.iso_runit D.comp_assoc
                     D.invert_side_of_triangle(2)
                       [of "(G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]" "D.inv (\<tau>\<^sub>1 a) \<cdot>\<^sub>D (\<tau>\<^sub>0 a \<star>\<^sub>D F.unit a)"
                           "\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]"]
               by simp
             thus ?thesis by simp
           qed
           also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a D.whisker_left by simp
           also have "... = ((\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a)"
             using a D.whisker_right by simp
           finally have "((\<tau>\<^sub>0' a \<star>\<^sub>D D.inv (\<tau>\<^sub>1 a)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                           = ((\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, G\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
           using a D.assoc_naturality [of "\<tau>\<^sub>0' a" "G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a" "\<tau>\<^sub>0' a"] D.comp_assoc
           by simp
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, G\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "(\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                   = \<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D ((G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)"
             using a D.whisker_left by simp
           also have "... = \<tau>\<^sub>0' a \<star>\<^sub>D (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D \<a>\<^sub>D[G\<^sub>0 a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]"
             using a D.assoc_naturality [of "G.unit a" "\<tau>\<^sub>0 a" "\<tau>\<^sub>0' a"] by simp
           also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a])"
             using a 2 D.whisker_left [of "\<tau>\<^sub>0' a"] by simp
           finally have "(\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' a \<star>\<^sub>D (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                           = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a])"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = ((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D G\<^sub>0 a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, G\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "(\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a)
                   = \<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<epsilon> a"
             using a D.whisker_left D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "G a" "G.unit a" "\<epsilon> a" "\<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a"]
             by simp
           also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D G\<^sub>0 a) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a)"
             using a 2 D.whisker_left D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "G.unit a" "G\<^sub>0 a" "G\<^sub>0 a" "\<epsilon> a"]
             by simp
           finally have "(\<tau>\<^sub>0' a \<star>\<^sub>D G a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a)
                           = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D G\<^sub>0 a) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, G\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "(\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D G\<^sub>0 a)
                   = \<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a] \<cdot>\<^sub>D (G.unit a \<star>\<^sub>D G\<^sub>0 a)"
             using a 2 D.whisker_left by simp
           also have "... = \<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<cdot>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]"
             using a D.runit_naturality [of "G.unit a"] by simp
           also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a])"
             using a 2 D.whisker_left by simp
           finally have "(\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a \<star>\<^sub>D G\<^sub>0 a)
                           = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a])"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' a, G\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          (\<r>\<^sub>D[\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
           using a 1 D.whisker_right D.runit_hcomp(1) D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          (\<r>\<^sub>D[\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
           using a D.assoc_naturality [of "\<tau>\<^sub>0' a" "\<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a]" "\<tau>\<^sub>0' a"] D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a])) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          (\<r>\<^sub>D[\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
           using a 2 D.whisker_left D.lunit_hcomp(4) D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          ((\<r>\<^sub>D[\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "(\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a])
                   = \<tau>\<^sub>0' a \<star>\<^sub>D (G\<^sub>0 a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a]"
             using a 2 D.whisker_left by simp
           also have "... = \<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a] \<cdot>\<^sub>D \<epsilon> a"
             using a D.lunit'_naturality [of "\<epsilon> a"] by simp
           also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<epsilon> a)"
             using a 2 D.whisker_left by simp
           finally have "(\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a])
                           = (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<epsilon> a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          (\<eta> a \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D
                          (\<r>\<^sub>D[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           have "(\<r>\<^sub>D[\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                   = \<r>\<^sub>D[\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<cdot>\<^sub>D (\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a 1 D.whisker_right by simp
           also have "... = \<eta> a \<cdot>\<^sub>D \<r>\<^sub>D[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a"
             using a D.runit_naturality [of "\<eta> a"] by simp
           also have "... = (\<eta> a \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (\<r>\<^sub>D[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
             using a 1 D.whisker_right by simp
           finally have "(\<r>\<^sub>D[\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D ((\<eta> a \<star>\<^sub>D F\<^sub>0 a) \<star>\<^sub>D \<tau>\<^sub>0' a)
                           = (\<eta> a \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (\<r>\<^sub>D[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a]) \<cdot>\<^sub>D
                          \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a]) \<cdot>\<^sub>D
                          \<l>\<^sub>D[\<tau>\<^sub>0' a] \<cdot>\<^sub>D
                          (\<r>\<^sub>D[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a)"
         proof -
           interpret adjoint_equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                       \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close>
             using a chosen_adjoint_equivalence by simp
           have "(\<tau>\<^sub>0' a \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' a, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (\<eta> a \<star>\<^sub>D \<tau>\<^sub>0' a)
                   = \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0' a]"
             using triangle_right by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0' a]"
         proof -
           have "\<l>\<^sub>D[\<tau>\<^sub>0' a] \<cdot>\<^sub>D (\<r>\<^sub>D[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a) = \<l>\<^sub>D[\<tau>\<^sub>0' a]"
           proof -
             have "D.seq \<r>\<^sub>D[F\<^sub>0 a] \<l>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 a]"
               using a 1 by auto
             thus ?thesis
               using a D.unitor_coincidence [of "F\<^sub>0 a"] D.comp_arr_dom D.whisker_right
               by (metis D.comp_arr_inv' D.iso_runit D.lunit_simps(1,4)
                   D.objE D.runit_simps(5) F.map\<^sub>0_simps(1) map\<^sub>0_simps(1,3))
           qed
           moreover have "(\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a]) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a]
                            = \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a]"
           proof -
             have "(\<tau>\<^sub>0' a \<star>\<^sub>D \<r>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a]) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a] =
                   ((\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D[G\<^sub>0 a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a])) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a]"
               using a D.unitor_coincidence D.comp_assoc by simp
             also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D \<l>\<^sub>D[G\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G\<^sub>0 a]) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a]"
               using a 2 D.whisker_left by simp
             also have "... = (\<tau>\<^sub>0' a \<star>\<^sub>D G\<^sub>0 a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a]"
               using a 2 D.comp_arr_inv' by simp
             also have "... = \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a]"
               using a 2 D.comp_cod_arr by simp
             finally show ?thesis by blast
           qed
           ultimately show ?thesis
             using D.comp_assoc by simp
         qed
         finally show "(\<tau>\<^sub>0' a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0' a]
                          = \<tau>\<^sub>1' a \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D \<tau>\<^sub>0' a)"
           by simp
       qed
       show "\<And>f g. \<lbrakk>C.ide f; C.ide g; src\<^sub>C g = trg\<^sub>C f\<rbrakk> \<Longrightarrow>
                      (\<tau>\<^sub>0' (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                      \<a>\<^sub>D[\<tau>\<^sub>0' (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D
                      (\<tau>\<^sub>1' g \<star>\<^sub>D G f) \<cdot>\<^sub>D
                      D.inv \<a>\<^sub>D[F g, \<tau>\<^sub>0' (src\<^sub>C g), G f] \<cdot>\<^sub>D
                      (F g \<star>\<^sub>D \<tau>\<^sub>1' f) \<cdot>\<^sub>D
                      \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' (src\<^sub>C f)]
                        = \<tau>\<^sub>1' (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D \<tau>\<^sub>0' (src\<^sub>C f))"
       proof -
         fix f g
         assume f: "C.ide f" and g: "C.ide g" and fg: "src\<^sub>C g = trg\<^sub>C f"
         let ?a = "src\<^sub>C f"
         let ?b = "trg\<^sub>C f"
         let ?c = "trg\<^sub>C g"
         (*
          * TODO: The following are extremely problematic.
          * I don't think all the cases are used, but they don't get discovered by auto
          * and sledgehammer doesn't find them, either.  Note that the same issue occurred
          * for the previous subgoal, where I needed D.ide (F\<^sub>0 a) and D.ide (G\<^sub>0 a).
          *)
         have *: "D.ide (G\<^sub>0 ?a) \<and> D.ide (G\<^sub>0 ?b) \<and> D.ide (G\<^sub>0 ?b) \<and> D.ide (G\<^sub>0 ?c)"
           using f g C.ideD(1) G.map\<^sub>0_simps(1) by blast
         have **: "D.ide (F\<^sub>0 ?a) \<and> D.ide (F\<^sub>0 ?b) \<and> D.ide (F\<^sub>0 ?b) \<and> D.ide (F\<^sub>0 ?c)"
           using f g C.ideD(1) F.map\<^sub>0_simps(1) by blast
         have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
               \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
               (\<tau>\<^sub>1' g \<star>\<^sub>D G f) \<cdot>\<^sub>D
               D.inv \<a>\<^sub>D[F g, \<tau>\<^sub>0' (src\<^sub>C g), G f] \<cdot>\<^sub>D
               (F g \<star>\<^sub>D \<tau>\<^sub>1' f) \<cdot>\<^sub>D
               \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]
                         = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                           \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                           ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D
                            \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                            ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                            (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b)
                               \<star>\<^sub>D G f) \<cdot>\<^sub>D
                            D.inv \<a>\<^sub>D[F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                            (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                                    (\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                                    (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                    \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                    ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                    (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                    ((\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                    (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
           unfolding map\<^sub>1_def
           using f g fg D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F g, \<tau>\<^sub>0' ?b, G f]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                 (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                 ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                 (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b)
                    \<star>\<^sub>D G f
                   = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f)"
           proof -
             have 1: "D.arr ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D
                             \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                             ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b))"
               using f g fg \<tau>.iso_map\<^sub>1_ide by auto
             thus ?thesis
               using f g fg D.whisker_right [of "G f"]
               by (metis D.seqE G.preserves_ide)
           qed
           moreover have "F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                                 (\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                                 (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                 \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                 ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                 (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                 ((\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                 (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                            = (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                              (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                              (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           proof -
             have 1: "D.arr ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a))"
               using f g fg \<tau>.iso_map\<^sub>1_ide map\<^sub>1_def map\<^sub>1_simps(1) by presburger
             thus ?thesis
               using f g fg D.whisker_left [of "F g"]
               by (metis D.seqE F.preserves_ide)
           qed
           ultimately show ?thesis
             using f g fg D.comp_assoc by simp
         qed
         (*
          * Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] and \<l>\<^sub>D\<^sup>-\<^sup>1[F f] to the bottom, followed by \<eta> ?c and D.inv (\<tau>\<^sub>1 g).
          * Move \<epsilon> ?b down across D.inv (\<tau>\<^sub>1 g) to meet \<eta> ?b, where they can be canceled.
          *)
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F g, \<tau>\<^sub>0' ?b, G f]
                   = \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f)"
             using f g fg D.assoc'_naturality [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "\<tau>\<^sub>0' ?b" "G f"] by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f])
                   = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F g" "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f" "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]"]
             by simp
           also have "... = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                            (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a"]
             by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f])
                           = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                   = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F g" "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"]
             by simp
           also have "... = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                              (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                         (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                           = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                   = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F g" "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
             by simp
           also have "... = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (F g \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                           = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                   = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F g" "\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
             by simp
           also have "... = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                           = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
          proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F g" "(\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "(\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (F g \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F g" "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (F g \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F g" "((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "(\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "(\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "(F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (F g \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D\<^sup>-\<^sup>1[F g] down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F g" "(F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?c down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          ((((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f)) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f]
                   = \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                     ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f)"
             using f g fg D.assoc'_naturality [of "\<eta> ?c \<star>\<^sub>D F g" "\<tau>\<^sub>0' ?b" "G f"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?c down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f) \<cdot>\<^sub>D ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f])
                   = (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<eta> ?c \<star>\<^sub>D F g" "F\<^sub>0 ?c \<star>\<^sub>D F g"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f" "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                            ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<eta> ?c \<star>\<^sub>D F g"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]" "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a"]
             by simp
           finally have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f])
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                             ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?c down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                 ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                   = (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<eta> ?c \<star>\<^sub>D F g" "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                            ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<eta> ?c \<star>\<^sub>D F g"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a" "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                         ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                             ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?c down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                   = (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<eta> ?c \<star>\<^sub>D F g" "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<eta> ?c \<star>\<^sub>D F g"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
                                     "\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?c down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                   = (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<eta> ?c \<star>\<^sub>D F g" "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<eta> ?c \<star>\<^sub>D F g"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?c down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<eta> ?c \<star>\<^sub>D F g" "F\<^sub>0 ?c \<star>\<^sub>D F g"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<eta> ?c \<star>\<^sub>D F g"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?c down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<eta> ?c \<star>\<^sub>D F g" "F\<^sub>0 ?c \<star>\<^sub>D F g"
                                     "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<eta> ?c \<star>\<^sub>D F g"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?c down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<eta> ?c \<star>\<^sub>D F g" "F\<^sub>0 ?c \<star>\<^sub>D F g"
                                     "((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "(\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f \<star>\<^sub>D G\<^sub>0 ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "\<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f] \<cdot>\<^sub>D
               (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<r>\<^sub>D[G f])
                  = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                    \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f \<star>\<^sub>D G\<^sub>0 ?a]"
             using f g fg D.assoc'_naturality [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<tau>\<^sub>0' ?b" "\<r>\<^sub>D[G f]"]
             by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "\<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f \<star>\<^sub>D G\<^sub>0 ?a] \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                  = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                    \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
            using f g fg D.assoc'_naturality [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<tau>\<^sub>0' ?b" "G f \<star>\<^sub>D \<epsilon> ?a"]
            by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "\<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                  = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                  \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
            using f g fg
                  D.assoc'_naturality [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<tau>\<^sub>0' ?b" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
            by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                    ((\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
          also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]"
            using f g fg \<tau>.iso_map\<^sub>1_ide
                  D.assoc_naturality [of "\<tau>\<^sub>0' ?b" "D.inv (\<tau>\<^sub>1 f)" "\<tau>\<^sub>0' ?a"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
          finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                        (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "\<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                    \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.assoc'_naturality
                    [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<tau>\<^sub>0' ?b" "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                  = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "G f" "\<r>\<^sub>D[G f]"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                           ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "\<r>\<^sub>D[G f]" "G f \<star>\<^sub>D G\<^sub>0 ?a"]
            by simp
          finally have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                        ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                            ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                  = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "G f \<star>\<^sub>D G\<^sub>0 ?a" "G f \<star>\<^sub>D \<epsilon> ?a"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                           ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "G f \<star>\<^sub>D \<epsilon> ?a" "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                        ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                            ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                  = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                           ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
                                    "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                  = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "G f" "\<r>\<^sub>D[G f]"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                           (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "\<r>\<^sub>D[G f]" "G f \<star>\<^sub>D G\<^sub>0 ?a"]
            by simp
          finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                        (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                  = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "G f \<star>\<^sub>D G\<^sub>0 ?a" "G f \<star>\<^sub>D \<epsilon> ?a"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                           (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "G f \<star>\<^sub>D \<epsilon> ?a" "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                        (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                  = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                           (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]" "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                    "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                  = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                    (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.assoc'_naturality [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]" "\<tau>\<^sub>0' ?b"
                                          "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])
                  = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g"
                                    "\<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]"]
            by simp
          also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                           (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g" "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]"
                                    "(\<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])
                          = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g"
                                    "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g" "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                  = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                    ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide
                  D.assoc'_naturality
                    [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)" "\<tau>\<^sub>0' ?b" "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)" "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g"
                                    "\<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]"]
            by simp
          also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                           ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b" "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]"
                                    "(\<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<tau>\<^sub>0' ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])
                          = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move D.inv (\<tau>\<^sub>1 g) down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)" "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g"
                                    "(\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b" "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)"
                                    "\<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move \<eta> ?b up. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<cdot>\<^sub>D (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D
                    (((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D ((\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg
                  D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]" "\<eta> ?c \<star>\<^sub>D F g"
                                    "((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                    "(\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<cdot>\<^sub>D (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D ((\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg D.comp_cod_arr by simp
          also have "... = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<cdot>\<^sub>D (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D
                           ((\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D ((F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg D.comp_arr_dom by simp
          also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           using f g fg ** D.interchange by auto
          finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move \<eta> ?b up. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D ((\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_cod_arr
                  D.interchange
                    [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)" "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]"
                        "((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom
                  D.interchange
                    [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)" "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]"
                        "(\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D ((\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move \<epsilon> ?b down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f"
            using f g fg D.whisker_right by simp
          also have "... = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D
                           (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f"
            using f g fg D.pentagon
                  D.invert_side_of_triangle(2)
                    [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                         "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                         "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b"]
                  D.comp_assoc
            by simp
          also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                           (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                           ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f)"
            using f g fg D.whisker_right by simp
          finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                        (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)
                          = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                            ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move \<epsilon> ?b down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f)) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f"
            using f g fg D.whisker_right [of "G f"] by simp
          also have "... = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f"
            using f g fg D.assoc_naturality [of "\<tau>\<^sub>0' ?c" "G g" "\<epsilon> ?b"] by simp
          also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f)"
            using f g fg * D.whisker_right [of "G f"] by simp
          finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                        (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)
                          = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Prepare to move \<epsilon> ?b down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D
                (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b)
                  = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
            using f g fg D.pentagon by simp
          hence "\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]
                   = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b)"
            using f g fg D.comp_assoc D.invert_side_of_triangle(1) by simp
          hence "\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b)
                   = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
            using f g fg D.comp_assoc
                  D.invert_side_of_triangle(2)
                    [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                        "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D
                           \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                        "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b"]
            by simp
          hence "\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f
                   = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f"
            using f g fg D.whisker_right by simp
          hence "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                 ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f)
                   = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                     (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)"
            using f g fg D.whisker_right by simp
         thus ?thesis
           using D.comp_assoc by simp
        qed
        (* Move \<epsilon> ?b down *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f)) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f)) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)
                  = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f"
            using f g fg D.whisker_right by simp
          also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f"
            using f g fg D.assoc'_naturality [of "\<tau>\<^sub>0' ?c" "G g" "\<epsilon> ?b"] by simp
          also have "... = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f)"
            using f g fg * D.whisker_right by simp
          finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                        (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f)
                          = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* There is a cancellation of associativities here. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f)
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f"
            using f g fg * D.comp_arr_inv' D.whisker_right
            by (metis C.ideD(1) C.obj_trg D.comp_assoc_assoc'(1) D.hcomp_simps(2) D.hseqI'
                D.ideD(1) G.map\<^sub>0_simps(3) G.preserves_ide G.preserves_src G.preserves_trg
                map\<^sub>0_simps(1-2))
          moreover have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f)
                           = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f"
            using f g fg D.comp_cod_arr by simp
          ultimately have "((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                           (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f)) \<cdot>\<^sub>D
                           ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f)
                             = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f"
            by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move \<r>\<^sub>D[G f] up. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                 (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D \<r>\<^sub>D[G f]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f" "\<r>\<^sub>D[G f]"]
             by simp
           also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "\<r>\<^sub>D[G f]" "G f \<star>\<^sub>D G\<^sub>0 ?a"]
             by simp
           finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                           = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             by blast
           thus ?thesis
             using f g fg D.comp_assoc by simp
         qed
         (* Move \<r>\<^sub>D[G f] up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D \<r>\<^sub>D[G f]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f" "\<r>\<^sub>D[G f]"]
             by simp
           also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "\<r>\<^sub>D[G f]" "G f \<star>\<^sub>D G\<^sub>0 ?a"]
             by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                           = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<r>\<^sub>D[G f] up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b" "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f" "\<r>\<^sub>D[G f]"]
             by simp
           also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b" "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b"
                                     "\<r>\<^sub>D[G f]" "G f \<star>\<^sub>D G\<^sub>0 ?a"]
             by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                           = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<epsilon> ?a up. The useful effect is on the associativity part of the term. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                 (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f \<star>\<^sub>D G\<^sub>0 ?a" "G f \<star>\<^sub>D \<epsilon> ?a"]
             by simp
           also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "G f \<star>\<^sub>D \<epsilon> ?a" "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                           = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<epsilon> ?a up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f \<star>\<^sub>D G\<^sub>0 ?a" "G f \<star>\<^sub>D \<epsilon> ?a"]
             by simp
           also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "G f \<star>\<^sub>D \<epsilon> ?a" "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                           = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<epsilon> ?a up and \<epsilon> ?b down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b"
                                     "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f \<star>\<^sub>D G\<^sub>0 ?a" "G f \<star>\<^sub>D \<epsilon> ?a"]
             by simp
           also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b"
                                     "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b"
                                     "G f \<star>\<^sub>D \<epsilon> ?a" "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                           = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Prepare to move \<epsilon> ?b down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           proof -
             have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D
                   (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b)
                     = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
               using f g fg D.pentagon by simp
             hence "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]
                      = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D
                        (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b)"
               using f g fg D.comp_assoc
                     D.invert_side_of_triangle(2)
                       [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                           "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                           "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b"]
               by simp
             hence "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]
                        \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a
                      = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<cdot>\<^sub>D
                        (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b)
                            \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"
               by simp
             thus ?thesis
               using f g fg D.whisker_right by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<epsilon> ?b down. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b]
                         \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.whisker_right by simp
           also have "... = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b)
                               \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.assoc_naturality [of "\<tau>\<^sub>0' ?c" "G g" "\<epsilon> ?b"] by simp
           also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg * D.whisker_right by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Prepare to move D.inv (\<tau>\<^sub>1 f) up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                   = ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
             by simp
           also have "... = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]" "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                           = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Prepare to move D.inv (\<tau>\<^sub>1 f) up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]" "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Prepare to move D.inv (\<tau>\<^sub>1 f) up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                   = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b"
                                     "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]" "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move D.inv (\<tau>\<^sub>1 f) up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move D.inv (\<tau>\<^sub>1 f) up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b]"
                                     "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move D.inv (\<tau>\<^sub>1 f) up across \<epsilon> ?b. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b"
                                     "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b"
                                     "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<eta> ?b up. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                   = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b" "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)"
                                     "(\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (*
          * Introduce associativities to align (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) with
          * (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) in the middle.
          *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                   = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
           proof -
             have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                     = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                       (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
               using f g fg D.comp_arr_dom by simp
             also have "... = ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                       = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                          ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                          ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                          ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                          ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                          \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]\<rbrace>"
                 using f g fg D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                       D.VV.ide_char D.VV.arr_char
                 by simp
               also have "... =
                          \<lbrace>((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<rbrace>"
                 using f g fg by (intro EV.eval_eqI, auto)
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                 using f g fg D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                       D.VV.ide_char D.VV.arr_char
                 by simp
               finally have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                               = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                 by blast
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                       = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg
                       D.assoc'_naturality [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g" "\<epsilon> ?b" "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                 by auto
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                       = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])"
                 using f g fg * D.comp_arr_dom D.comp_cod_arr D.whisker_left
                       D.interchange [of "\<epsilon> ?b" "(\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b)" "(\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                         "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]"]
                 by simp
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg * D.comp_arr_dom D.comp_cod_arr D.whisker_left
                       D.interchange [of "G\<^sub>0 ?b" "\<epsilon> ?b" "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]"
                                         "\<tau>\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                 by simp
               finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                               = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 by blast
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])
                       = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D
                           (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           \<a>\<^sub>D[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
                 using f g fg D.whisker_left by simp
               also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D
                                  \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                  ((\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg D.assoc_naturality [of "\<epsilon> ?b" "\<tau>\<^sub>0 ?b" "F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                 by simp
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg * D.whisker_left by simp
               finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])
                               = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 by blast
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             finally show ?thesis by simp
           qed
           moreover
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                   = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])"
           proof -
             have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                     = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                       ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
               using f g fg ** D.comp_arr_dom by simp
             also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])"
 
             proof -
               have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                       = \<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                          \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                          ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                          ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                          \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                          (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>])\<rbrace>"
                 using f g fg ** D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                       D.VV.ide_char D.VV.arr_char
                 by simp
               also have "... = \<lbrace>(\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<rbrace>"
                 using f g fg by (intro EV.eval_eqI, auto)
               also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                 using f g fg ** D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                       D.VV.ide_char D.VV.arr_char
                 by simp
               finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                             (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                               = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                 by blast
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])"
             proof -
               have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                     = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D
                         ((\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]"
                 using f g fg ** D.comp_arr_dom D.comp_cod_arr
                       D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b" "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b]"
                                         "(\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]"]
                 by simp
               also have "... = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D
                                  \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                  (\<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg ** D.assoc'_naturality [of "\<eta> ?b" "F f" "\<tau>\<^sub>0' ?a"]
                 by simp
               also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg ** D.comp_arr_dom D.comp_cod_arr
                       D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b]" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b"
                                         "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]"
                                         "\<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                 by simp
               finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D (\<eta> ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                               = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                 (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 by blast
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])"
             proof -
               have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                       = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg D.assoc'_naturality [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g" "\<tau>\<^sub>0 ?b"
                                                      "\<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                 by force
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])"
             proof -
               have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])
                       = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D
                           (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
                 using f g fg ** D.whisker_left by auto
               also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D
                                   \<a>\<^sub>D[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                   ((\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg ** D.assoc_naturality [of "\<tau>\<^sub>0 ?b" "\<eta> ?b" "F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                 by auto
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg ** D.whisker_left by auto
               finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])
                               = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 by blast
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             finally show ?thesis by blast
           qed
           ultimately show ?thesis
             using D.comp_assoc by simp
         qed
         (* Cancel out all the intervening associativities. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"
           proof -
             have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   (\<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b] \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                   ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<tau>\<^sub>0' ?b) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                   \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                   (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (\<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a])
                     = \<lbrace>((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        (\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>) \<^bold>\<cdot>
                        ((\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>) \<^bold>\<cdot>
                        \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>) \<^bold>\<cdot>
                        (\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                         \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, (\<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>])\<rbrace>"
               using f g fg D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                     D.VV.ide_char D.VV.arr_char
               by simp
             also have "... =
                        \<lbrace>(\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<rbrace>"
               using f g fg by (intro EV.eval_eqI, auto)
             also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"
               using f g fg D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                     D.VV.ide_char D.VV.arr_char
               by simp
             finally show ?thesis
               using D.comp_assoc by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Hooray!  We can finally cancel \<eta> ?b with \<epsilon> ?b using the triangle identity. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 ?b] \<star>\<^sub>D (F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D
                       ((\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                       (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                       ((\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.whisker_left by simp
           also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D
                               (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b] \<cdot>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b)
                                  \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.whisker_right by simp
           also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 ?b] \<star>\<^sub>D (F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           proof -
             interpret adjoint_equivalence_in_bicategory
                          V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 ?b\<close> \<open>\<tau>\<^sub>0' ?b\<close> \<open>\<eta> ?b\<close> \<open>\<epsilon> ?b\<close>
               using f chosen_adjoint_equivalence by simp
             show ?thesis
               using fg triangle_left by simp
           qed
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<epsilon> ?b \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, \<tau>\<^sub>0' ?b, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D \<eta> ?b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 ?b] \<star>\<^sub>D (F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Simplify some more canonical arrows. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 ?b] \<star>\<^sub>D (F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                   = (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           proof -
             have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, G\<^sub>0 ?b, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G\<^sub>0 ?b, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 ?b] \<star>\<^sub>D (F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F\<^sub>0 ?b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                     = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>G\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0, (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>G\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>G\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<r>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> (\<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>)) \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>])\<rbrace>"
               using f g fg * ** D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                     D.VV.ide_char D.VV.arr_char D.\<ll>_ide_simp D.\<rr>_ide_simp
               by simp
             also have "... = \<lbrace>(\<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                               ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<l>\<^bold>[\<^bold>\<langle>F f\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>) \<^bold>\<cdot>
                               \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, (\<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                               (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> (\<^bold>\<langle>F\<^sub>0 ?b\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>)\<rbrace>"
               using f g fg by (intro EV.eval_eqI, auto)
             also have "... = (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
               using f g fg * ** D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                     D.VV.ide_char D.VV.arr_char D.\<ll>_ide_simp D.\<rr>_ide_simp
               by simp
             finally show ?thesis by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D G f) \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<r>\<^sub>D[G f])
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]" "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b" "G f" "\<r>\<^sub>D[G f]"]
             by simp
           moreover have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                            = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
             using f g fg * D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b]" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b"
                                     "G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
             by simp
           ultimately show ?thesis
             using D.comp_assoc by simp
         qed
         (*
          * Move \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] and \<l>\<^sub>D[F f] outside, to get rid of
          * G\<^sub>0 ?b and F\<^sub>0 ?b.
          *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.assoc_naturality [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g" "\<tau>\<^sub>0 ?b" "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b" "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b]"
                                      "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b]" "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b"
                                     "F f \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b" "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)"
                                     "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)" "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g"
                                     "F f \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<tau>\<^sub>0 ?b) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                   = \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b" "\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g]"
                                     "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]"]
             by simp
           also have "... = (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g]" "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g" "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
                                     "(D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]"]
                   D.whisker_left
             by simp (* 12 sec *)
           finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                           = (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Introduce associativities to achieve (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) and (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)). *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                   = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
           proof -
             have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                     = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                       ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
               using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom by simp
             also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                       = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                   \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left D.comp_assoc by simp
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_inv' by simp
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_cod_arr by simp
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_inv_arr' by simp
               finally show ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                       = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg \<tau>.iso_map\<^sub>1_ide
                       D.assoc'_naturality [of "\<tau>\<^sub>0' ?c" "G g" "D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                 by simp
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])
                       = \<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
               also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                         ((G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.assoc_naturality [of "G g" "D.inv (\<tau>\<^sub>1 f)" "\<tau>\<^sub>0' ?a"]
                 by simp
               also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
               finally have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a])
                               = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 by blast
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             finally show ?thesis by simp
           qed
           moreover have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a
                            = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
           proof -
             have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a
                     = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                       ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
               using f g fg \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom by auto
             also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                     \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                       = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                   \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
                 using f g fg D.whisker_left D.comp_assoc by simp
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                (\<tau>\<^sub>0' ?c \<star>\<^sub>D (\<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
                 using f g fg D.comp_arr_inv' by simp
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
                 using f g fg D.comp_cod_arr by simp
               also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                                ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg D.comp_inv_arr' by simp
               finally show ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g)) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                       = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg \<tau>.iso_map\<^sub>1_ide
                       D.assoc'_naturality [of "\<tau>\<^sub>0' ?c" "D.inv (\<tau>\<^sub>1 g)" "F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                 by simp
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                              \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
             proof -
               have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                     (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a])
                       = \<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
               also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                         ((D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.assoc_naturality [of "D.inv (\<tau>\<^sub>1 g)" "F f" "\<tau>\<^sub>0' ?a"]
                 by simp
               also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
               finally have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a])
                               = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
                 by blast
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             finally show ?thesis
               by simp
           qed
           ultimately show ?thesis
             using D.comp_assoc by simp
         qed
         (* Cancel intervening associativities. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                   = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
           proof -
             have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, (\<tau>\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g, \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, \<tau>\<^sub>0 ?b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                   \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D \<tau>\<^sub>0 ?b, F f, \<tau>\<^sub>0' ?a])
                     = \<lbrace>(\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle>, (\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        ((\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G g\<^bold>\<rangle>,\<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>) \<^bold>\<cdot>
                        \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>G g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        (\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>G g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>])\<rbrace>"
               using f g fg D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                     D.VV.ide_char D.VV.arr_char
               by simp
             also have "... = \<lbrace>\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>G g\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<rbrace>"
               using f g fg by (intro EV.eval_eqI, auto)
             also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
               using f g fg D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                     D.VV.ide_char D.VV.arr_char
               by simp
             finally show ?thesis by blast
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Apply "respects composition". *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<tau>\<^sub>0' ?c \<star>\<^sub>D ((G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (\<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              ((D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left by simp
           also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D
                              (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f)
                                \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_right by simp
           also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D
                                \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<cdot>\<^sub>D
                                (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D
                                D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D
                                ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f])
                                   \<star>\<^sub>D \<tau>\<^sub>0' ?a"
           proof -
             have 1: "(\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f)
                        = D.inv ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f]) \<cdot>\<^sub>D
                          \<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a]"
             proof -
               have "((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<cdot>\<^sub>D
                     (\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f)) \<cdot>\<^sub>D
                     \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a]
                       = \<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a)"
                 using f g fg \<tau>.respects_hcomp D.comp_assoc by simp
               hence "((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f]) \<cdot>\<^sub>D
                      (\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f)
                        = \<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a]"
                 using f g fg D.comp_assoc
                       D.invert_side_of_triangle(2)
                         [of "\<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a)"
                             "(\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<cdot>\<^sub>D
                              (\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f)"
                             "\<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a]"]
                 by fastforce
               moreover have "D.iso ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f])"
                 using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_simp F.cmp_components_are_iso
                       F.FF_def
                 by (intro D.isos_compose D.seqI D.hseqI') auto
               moreover have "D.seq (\<tau>\<^sub>1 (g \<star>\<^sub>C f))
                                    ((\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a])"
                 using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_simp G.FF_def
                 by (intro D.seqI) auto
               ultimately show ?thesis
                 using f g fg \<tau>.iso_map\<^sub>1_ide
                       D.invert_side_of_triangle(1)
                         [of "\<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a]"
                             "(\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f]"
                             "(\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f)"]
                 by simp
             qed
             have "(G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f)
                     = D.inv ((\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f))"
             proof -
               have "D.inv ((\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f))
                       = D.inv (\<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f)) \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 g \<star>\<^sub>D F f)"
               proof -
                 have "D.iso ((\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f))"
                   using f g fg \<tau>.iso_map\<^sub>1_ide
                   by (intro D.isos_compose) auto
                 moreover have "D.iso (\<tau>\<^sub>1 g \<star>\<^sub>D F f)"
                   using f g fg \<tau>.iso_map\<^sub>1_ide by auto
                 ultimately show ?thesis
                   using \<tau>.iso_map\<^sub>1_ide D.inv_comp_left by simp
               qed
               also have "... = ((G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f]) \<cdot>\<^sub>D
                                (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f)"
               proof -
                 have "D.iso (\<a>\<^sub>D\<^sup>-\<^sup>1[G g, \<tau>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f))"
                   using f g fg \<tau>.iso_map\<^sub>1_ide
                   by (intro D.isos_compose) auto
                 thus ?thesis
                   using f g fg \<tau>.iso_map\<^sub>1_ide D.inv_comp_left by simp
               qed
               finally show ?thesis
                 using D.comp_assoc by simp
             qed
             also have "... = D.inv (D.inv ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f]) \<cdot>\<^sub>D
                                     \<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D
                                     (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a])"
               using 1 by simp
             also have "... = D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D
                                     \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a]) \<cdot>\<^sub>D
                              ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f])"
             proof -
               have 2: "D.iso ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f])"
                  using f g fg \<tau>.iso_map\<^sub>1_ide C.VV.arr_char C.VV.dom_simp F.cmp_components_are_iso
                        F.FF_def
                  by (intro D.isos_compose) auto
               moreover have "D.iso (D.inv ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f]) \<cdot>\<^sub>D
                                     \<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a])"
                  using 2 f g fg \<tau>.iso_map\<^sub>1_ide C.VV.arr_char C.VV.dom_simp
                        C.VV.cod_simp F.FF_def G.FF_def D.inv_comp_left
                  by (intro D.isos_compose) auto (* 10 sec *)
               ultimately show ?thesis
                 using D.inv_comp_left
                         [of "D.inv ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f])"
                             "\<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a]"]
                       D.inv_inv D.iso_inv_iso
                 by metis
             qed
             also have "... = \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<cdot>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D
                              D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f])"
             proof -
               have "D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a])
                       = \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<cdot>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D
                         D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f))"
               proof -
                 have 2: "D.iso (\<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a])"
                    using f g fg \<tau>.iso_map\<^sub>1_ide C.VV.arr_char C.VV.dom_simp
                          C.VV.cod_simp G.FF_def
                    by (intro D.isos_compose D.seqI D.hseqI') auto
                 have "D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a])
                         = D.inv ((\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G g, G f, \<tau>\<^sub>0 ?a]) \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f))"
                   using 2 f g fg * \<tau>.iso_map\<^sub>1_ide G.cmp_components_are_iso C.VV.arr_char
                         C.VV.dom_simp C.VV.cod_simp G.FF_def D.inv_comp_left
                   by simp
                 also have "... = (\<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<cdot>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a)) \<cdot>\<^sub>D
                                  D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f))"
                   using 2 f g fg * \<tau>.iso_map\<^sub>1_ide G.cmp_components_are_iso C.VV.arr_char
                         C.VV.dom_simp C.VV.cod_simp G.FF_def D.inv_comp_left
                   by simp
                 also have "... = \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<cdot>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D
                                  D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f))"
                   using D.comp_assoc by simp
                 finally show ?thesis
                   using D.comp_assoc by simp
               qed
               thus ?thesis
                 using D.comp_assoc by simp
             qed
             finally have "(G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f)
                             = \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<cdot>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D
                               D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<cdot>\<^sub>D (\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f]"
               by blast
             thus ?thesis by simp
           qed
           also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D
                              (\<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              ((D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                              (((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                              (\<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_right
                   C.VV.arr_char C.VV.dom_simp C.VV.cod_simp G.cmp_components_are_iso
                   F.FF_def G.FF_def
             by simp
           also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide D.whisker_left
                   C.VV.arr_char C.VV.dom_simp C.VV.cod_simp G.cmp_components_are_iso
                   F.FF_def G.FF_def
             by simp
           finally have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, \<tau>\<^sub>0 ?b, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<tau>\<^sub>1 g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move \<l>\<^sub>D[F f] down, where it will cancel. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g" "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]"
                                     "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                   D.whisker_right
             by simp
           also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]" "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g"
                                     "F f \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
                   D.whisker_right
             by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
          thus ?thesis
             using D.comp_assoc by simp
        qed
        (* Move \<l>\<^sub>D[F f] down. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = (\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g" "\<eta> ?c \<star>\<^sub>D F g"
                                    "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a" "(F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg D.comp_arr_dom D.comp_cod_arr
                  D.interchange [of "\<eta> ?c \<star>\<^sub>D F g" "F\<^sub>0 ?c \<star>\<^sub>D F g" "F f \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D (F\<^sub>0 ?b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Cancel \<l>\<^sub>D[F f] and \<l>\<^sub>D\<^sup>-\<^sup>1[F f]. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) =
                \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D (\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg D.comp_cod_arr
                  D.interchange [of "F\<^sub>0 ?c \<star>\<^sub>D F g" "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "\<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a" "\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D[F f] \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
            using f g fg D.whisker_right by simp
          also have "... = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"
            using f D.comp_arr_inv' by simp
          finally have "((F\<^sub>0 ?c \<star>\<^sub>D F g) \<star>\<^sub>D \<l>\<^sub>D[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                        (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move \<r>\<^sub>D\<^sup>-\<^sup>1[G g] up, where it will cancel. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
            using f g fg * D.comp_arr_dom
                  D.interchange [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b]" "\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g]"
                                    "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]" "(G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
            by simp
          also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
            using f g fg D.runit_hcomp(4) [of "\<tau>\<^sub>0' ?c" "G g"] by simp
          finally have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G\<^sub>0 ?b] \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                        (\<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D G g] \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move \<r>\<^sub>D\<^sup>-\<^sup>1[G g] up. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D (G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
            using f g fg D.comp_cod_arr
                  D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b" "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]"
                                    "G f \<star>\<^sub>D \<epsilon> ?a" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
            by simp
          also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                           ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
            using f g fg D.comp_arr_dom
                  D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]" "\<tau>\<^sub>0' ?c \<star>\<^sub>D G g" "G f \<star>\<^sub>D \<epsilon> ?a"
                                    "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
            by simp
          finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G\<^sub>0 ?b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                        ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])
                          = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Cancel \<r>\<^sub>D\<^sup>-\<^sup>1[G g] and \<r>\<^sub>D[G g]. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f] \<cdot>\<^sub>D (G f \<star>\<^sub>D \<epsilon> ?a)"
            using f g fg D.interchange by simp
          also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g] \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f] \<cdot>\<^sub>D (G f \<star>\<^sub>D \<epsilon> ?a)"
            using f g fg D.whisker_left by simp
          also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<r>\<^sub>D[G f] \<cdot>\<^sub>D (G f \<star>\<^sub>D \<epsilon> ?a)"
            using g D.comp_arr_inv' by simp
          also have "... = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)"
            using f g fg D.whisker_left by simp
          finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g]) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                        ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G g]) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)
                          = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, (G f \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                  = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                    (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
            using f g fg
                  D.assoc'_naturality [of "\<tau>\<^sub>0' ?c" "G g" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"]
            by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = \<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                             \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                             (\<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg D.whisker_left by simp
          also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                                    \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
            using f g fg D.pentagon by simp
          also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                           (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
            using f g fg D.whisker_left by simp
          finally have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                        (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                        (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         (((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f \<star>\<^sub>D G\<^sub>0 ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]
                  = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f \<star>\<^sub>D G\<^sub>0 ?a] \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a)"
            using f g fg D.assoc'_naturality [of "\<tau>\<^sub>0' ?c" "G g" "G f \<star>\<^sub>D \<epsilon> ?a"]
            by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Move associativity to where it can be canceled. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D
                         \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f \<star>\<^sub>D G\<^sub>0 ?a]
                  = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<r>\<^sub>D[G f])"
            using f g fg D.assoc'_naturality [of "\<tau>\<^sub>0' ?c" "G g" "\<r>\<^sub>D[G f]"] by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Cancel associativities. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a])) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c, G g, G f]) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<r>\<^sub>D[G f])
                  = (\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<r>\<^sub>D[G f])"
            using f g fg D.comp_arr_inv' D.comp_cod_arr by simp
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Permute associativity to continue forming G g \<star>\<^sub>D G f. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, G\<^sub>0 ?a])) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a])
                  = \<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a]"
            using f g fg D.whisker_left by simp
          also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, G\<^sub>0 ?a] \<cdot>\<^sub>D ((G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a)"
            using f g fg D.assoc_naturality [of "G g" "G f" "\<epsilon> ?a"] by simp
          also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, G\<^sub>0 ?a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a)"
            using f g fg * D.whisker_left by simp
          finally have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D G g \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                        (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a])
                          = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g, G f, G\<^sub>0 ?a]) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a)"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
        qed
        (* Form \<r>\<^sub>D[G g \<star>\<^sub>D G f]. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g \<star>\<^sub>D G f]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
          using f g fg * D.runit_hcomp D.whisker_left D.comp_assoc by simp
        (* Move D.inv (\<Phi>\<^sub>G (g, f)) to the top and cancel with its inverse. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g \<star>\<^sub>D G f]) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
        proof -
          have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                  = \<tau>\<^sub>0' ?c \<star>\<^sub>D
                       \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                       ((D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
            using f g fg G.cmp_components_are_iso D.whisker_left by simp
          also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D
                             (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]"
            using f g fg G.cmp_components_are_iso
                  D.assoc_naturality [of "D.inv (\<Phi>\<^sub>G (g, f))" "\<tau>\<^sub>0 ?a" "\<tau>\<^sub>0' ?a"]
            by simp
          also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                           (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
            using f g fg G.cmp_components_are_iso D.whisker_left by simp
          finally have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G g \<star>\<^sub>D G f, \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                        (\<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                          = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a])"
            by blast
          thus ?thesis
            using D.comp_assoc by simp
         qed
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g \<star>\<^sub>D G f]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D G\<^sub>0 ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<tau>\<^sub>0' ?c \<star>\<^sub>D ((G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg G.cmp_components_are_iso D.whisker_left by simp
           also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<epsilon> ?a)"
             using f g fg * D.comp_arr_dom D.comp_cod_arr
                    D.interchange [of "G g \<star>\<^sub>D G f" "D.inv (\<Phi>\<^sub>G (g, f))" "\<epsilon> ?a" "\<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a"]
             by simp
           also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D (G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a)"
             using f g fg * D.comp_arr_dom D.comp_cod_arr
                   D.interchange [of "D.inv (\<Phi>\<^sub>G (g, f))" "G (g \<star>\<^sub>C f)" "G\<^sub>0 ?a" "\<epsilon> ?a"]
             by simp
           also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                            (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a)"
             using f g fg * G.cmp_components_are_iso D.whisker_left by simp
           finally have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D (G g \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D \<tau>\<^sub>0 ?a \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D G\<^sub>0 ?a) \<cdot>\<^sub>D
                             (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
        also have "... = (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)))) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)])) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g \<star>\<^sub>D G f]) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D G\<^sub>0 ?a)
                   = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g \<star>\<^sub>D G f] \<cdot>\<^sub>D (D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D G\<^sub>0 ?a)"
             using f g fg * G.cmp_components_are_iso D.whisker_left by simp
           also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]"
             using f g fg * G.cmp_components_are_iso D.runit_naturality [of "D.inv (\<Phi>\<^sub>G (g, f))"]
             by simp
           also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D  \<r>\<^sub>D[G (g \<star>\<^sub>C f)])"
             using f g fg * G.cmp_components_are_iso D.whisker_left by simp
           finally have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G g \<star>\<^sub>D G f]) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)) \<star>\<^sub>D G\<^sub>0 ?a) =
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D  \<r>\<^sub>D[G (g \<star>\<^sub>C f)])"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)))) \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)])
                   = \<tau>\<^sub>0' ?c \<star>\<^sub>D (\<Phi>\<^sub>G (g, f) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f))) \<cdot>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]"
             using f g fg D.whisker_left G.cmp_components_are_iso
                   C.VV.arr_char C.VV.dom_simp C.VV.cod_simp
             by simp
           also have "... = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]"
             using f g fg G.cmp_components_are_iso D.comp_arr_inv' D.comp_cod_arr by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<Phi>\<^sub>G (g, f)))) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)])
                           = \<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Perform similar manipulations on the bottom part. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[F\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           using f g fg D.assoc_naturality [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F g]" "F f" "\<tau>\<^sub>0' ?a"] D.comp_assoc by simp
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D \<a>\<^sub>D[F\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]
                   = \<a>\<^sub>D[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg **
                   D.assoc_naturality [of "\<eta> ?c \<star>\<^sub>D F g" "F f" "\<tau>\<^sub>0' ?a"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 \<a>\<^sub>D[(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg D.assoc_naturality [of "\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g]" "F f" "\<tau>\<^sub>0' ?a"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Replace \<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f by \<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)]. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D (F g), F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                    = \<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D (F g), F g, F f] \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.lunit_hcomp(2) [of "F g" "F f"] by simp
           also have "... = (\<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D (F g), F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg ** D.whisker_right [of "\<tau>\<^sub>0' ?a"] by simp
           finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a
                           = (\<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D (F g), F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Move the associativity up to the big block of associativities. *)
        also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                         ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D (F g), F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = ((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D (F g), F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg ** D.whisker_right by simp
           also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c, F g, F f] \<cdot>\<^sub>D (\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg ** D.assoc'_naturality [of "\<eta> ?c" "F g" "F f"] by simp
           also have "... = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            ((\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg ** D.whisker_right by simp    
           finally have "(((\<eta> ?c \<star>\<^sub>D F g) \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D (F g), F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             ((\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Simplify the block of associativities. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                 ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           proof -
             have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                   (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f \<star>\<^sub>D \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D F g, F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                   ((\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g] \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                   (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c, F g, F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                     = \<lbrace>(\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?c\<^bold>\<rangle>, \<^bold>\<langle>F g\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>) \<^bold>\<cdot>
                        (\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F g\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                        \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F g\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F g\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                        ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?c\<^bold>\<rangle>, \<^bold>\<langle>F g\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>) \<^bold>\<cdot>
                        (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 ?c\<^bold>\<rangle>, \<^bold>\<langle>F g\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>)\<rbrace>"
               using f g fg D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                     D.VV.ide_char D.VV.arr_char
               by simp
             also have "... = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>,  \<^bold>\<langle>\<tau>\<^sub>0 ?c\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                               (\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' ?c\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 ?c\<^bold>\<rangle>, \<^bold>\<langle>F g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' ?a\<^bold>\<rangle>)\<rbrace>"
               using f g fg by (intro EV.eval_eqI, auto)
             also have "... = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                              (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
               using f g fg D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char
                     D.VV.ide_char D.VV.arr_char
               by simp
             finally show ?thesis by simp
           qed
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* Permute \<Phi>\<^sub>F (g, f) to the bottom. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F (g \<star>\<^sub>C f), \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D ((\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F g \<star>\<^sub>D F f, \<tau>\<^sub>0' ?a]
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F (g \<star>\<^sub>C f), \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def
                   D.assoc_naturality [of "\<tau>\<^sub>0' ?c" "\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)" "\<tau>\<^sub>0' ?a"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F (g \<star>\<^sub>C f), \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D (\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f))) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def
                   D.whisker_right
             by simp
           also have "... = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D
                            ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def
                   D.assoc_naturality [of "\<tau>\<^sub>0' ?c" "\<tau>\<^sub>0 ?c" "\<Phi>\<^sub>F (g, f)"]
             by simp
           also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                            (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def
                   D.whisker_right
             by simp
           finally have "((\<tau>\<^sub>0' ?c \<star>\<^sub>D (\<tau>\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f))) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                             (((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F (g \<star>\<^sub>C f), \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((F\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 ((\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = ((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D (\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def D.whisker_right
             by simp
           also have "... = (\<eta> ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def
                   D.interchange [of "\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c" "\<eta> ?c" "\<Phi>\<^sub>F (g, f)" "F g \<star>\<^sub>D F f"]
             by simp
           also have "... = (\<eta> ?c \<star>\<^sub>D F (g \<star>\<^sub>C f)) \<cdot>\<^sub>D (F\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg D.comp_arr_dom D.comp_cod_arr
                   C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def
                   D.interchange [of "\<eta> ?c" "F\<^sub>0 ?c" "F (g \<star>\<^sub>C f)" "\<Phi>\<^sub>F (g, f)"]
             by simp
           also have "... = ((\<eta> ?c \<star>\<^sub>D F (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D ((F\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg ** C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def D.whisker_right
             by simp
           finally have "(((\<tau>\<^sub>0' ?c \<star>\<^sub>D \<tau>\<^sub>0 ?c) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                         ((\<eta> ?c \<star>\<^sub>D F g \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = ((\<eta> ?c \<star>\<^sub>D F (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D ((F\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F (g \<star>\<^sub>C f), \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "((F\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                   = (F\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg ** C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def D.whisker_right
             by simp
           also have "... = \<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<cdot>\<^sub>D \<Phi>\<^sub>F (g, f) \<star>\<^sub>D \<tau>\<^sub>0' ?a"
             using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def
                   D.lunit'_naturality [of "\<Phi>\<^sub>F (g, f)"]
             by simp
           also have "... = (\<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg C.VV.arr_char C.VV.dom_simp C.VV.cod_char F.FF_def D.whisker_right
             by simp
           finally have "((F\<^sub>0 ?c \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F g \<star>\<^sub>D F f] \<star>\<^sub>D \<tau>\<^sub>0' ?a)
                           = (\<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             by blast
           thus ?thesis
             using D.comp_assoc by simp
         qed
         (* One more associativity to move to its final position. *)
         also have "... = (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<r>\<^sub>D[G (g \<star>\<^sub>C f)]) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D G (g \<star>\<^sub>C f) \<star>\<^sub>D \<epsilon> ?a) \<cdot>\<^sub>D
                          (\<tau>\<^sub>0' ?c \<star>\<^sub>D \<a>\<^sub>D[G (g \<star>\<^sub>C f), \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a]) \<cdot>\<^sub>D
                          \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G (g \<star>\<^sub>C f) \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                          ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f))) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c, F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          ((\<eta> ?c \<star>\<^sub>D F (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<l>\<^sub>D\<^sup>-\<^sup>1[F (g \<star>\<^sub>C f)] \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                          (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
         proof -
           have "(\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f)) \<star>\<^sub>D \<tau>\<^sub>0' ?a) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<tau>\<^sub>0' ?c, \<tau>\<^sub>0 ?c \<star>\<^sub>D F (g \<star>\<^sub>C f), \<tau>\<^sub>0' ?a]
                   = \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G (g \<star>\<^sub>C f) \<star>\<^sub>D \<tau>\<^sub>0 ?a, \<tau>\<^sub>0' ?a] \<cdot>\<^sub>D
                     ((\<tau>\<^sub>0' ?c \<star>\<^sub>D D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f))) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
             using f g fg \<tau>.iso_map\<^sub>1_ide
                   D.assoc_naturality [of "\<tau>\<^sub>0' ?c" "D.inv (\<tau>\<^sub>1 (g \<star>\<^sub>C f))" "\<tau>\<^sub>0' ?a"]
             by simp
           thus ?thesis
             using D.comp_assoc by simp
         qed
         also have "... = \<tau>\<^sub>1' (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           unfolding map\<^sub>1_def
           using f g fg D.comp_assoc by simp
         finally show "(\<tau>\<^sub>0' ?c \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' ?c, G g, G f] \<cdot>\<^sub>D (\<tau>\<^sub>1' g \<star>\<^sub>D G f) \<cdot>\<^sub>D
                       D.inv \<a>\<^sub>D[F g, \<tau>\<^sub>0' (src\<^sub>C g), G f] \<cdot>\<^sub>D (F g \<star>\<^sub>D \<tau>\<^sub>1' f) \<cdot>\<^sub>D \<a>\<^sub>D[F g, F f, \<tau>\<^sub>0' ?a]
                         = \<tau>\<^sub>1' (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>F (g, f) \<star>\<^sub>D \<tau>\<^sub>0' ?a)"
           by blast
       qed
     qed
 
     lemma is_pseudonatural_equivalence:
     shows "pseudonatural_equivalence
              V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G F \<Phi>\<^sub>F \<tau>\<^sub>0' \<tau>\<^sub>1'"
       ..
 
   end
 
   subsection "Pseudonaturally Equivalent Pseudofunctors"
 
   text \<open>
     Pseudofunctors \<open>F\<close> and \<open>G\<close> are \emph{pseudonaturally equivalent} if there is a
     pseudonatural equivalence between them.
   \<close>
 
   definition pseudonaturally_equivalent
   where "pseudonaturally_equivalent
            V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<equiv>
          \<exists>\<tau>\<^sub>0 \<tau>\<^sub>1. pseudonatural_equivalence
                   V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
 
   lemma pseudonaturally_equivalent_reflexive:
   assumes "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
   shows "pseudonaturally_equivalent
            V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F F \<Phi>\<^sub>F"
   proof -
     interpret F: pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F
       using assms by simp
     interpret identity_pseudonatural_transformation
                 V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F
       ..
     show ?thesis
       using pseudonatural_equivalence_axioms pseudonaturally_equivalent_def by blast
   qed
 
   lemma pseudonaturally_equivalent_symmetric:
   assumes "pseudonaturally_equivalent
              V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G"
   shows "pseudonaturally_equivalent
              V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G F \<Phi>\<^sub>F"
   proof -
     obtain \<tau>\<^sub>0 \<tau>\<^sub>1 where \<tau>: "pseudonatural_equivalence
                            V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
       using assms pseudonaturally_equivalent_def by blast
     interpret \<tau>: pseudonatural_equivalence
                     V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
       using \<tau> by simp
     interpret \<sigma>: converse_pseudonatural_equivalence
                     V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
       ..
     show ?thesis
       using \<sigma>.is_pseudonatural_equivalence pseudonaturally_equivalent_def by blast
   qed
 
   lemma pseudonaturally_equivalent_transitive:
   assumes "pseudonaturally_equivalent
              V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G"
   and "pseudonaturally_equivalent
          V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G H \<Phi>\<^sub>H"
   shows "pseudonaturally_equivalent
            V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F H \<Phi>\<^sub>H"
   proof -
     obtain \<tau>\<^sub>0 \<tau>\<^sub>1 where \<tau>: "pseudonatural_equivalence
                             V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
       using assms pseudonaturally_equivalent_def by blast
     interpret \<tau>: pseudonatural_equivalence
                    V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
       using \<tau> by simp
     obtain \<sigma>\<^sub>0 \<sigma>\<^sub>1 where \<sigma>: "pseudonatural_equivalence
                              V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1"
       using assms pseudonaturally_equivalent_def by blast
     interpret \<sigma>: pseudonatural_equivalence
                    V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1
       using \<sigma> by simp
     interpret \<sigma>\<tau>: composite_pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
                     V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1 \<sigma>\<^sub>0 \<sigma>\<^sub>1
       ..
     show ?thesis
       using \<sigma>\<tau>.pseudonatural_equivalence_axioms pseudonaturally_equivalent_def by blast
   qed
 
   text \<open>
     If \<open>\<tau>\<close> is a pseudonatural equivalence from pseudofunctor \<open>F\<close> to pseudofunctor \<open>G\<close>
     and \<open>\<sigma>\<close> is the converse equivalence, then \<open>F\<close> is locally naturally isomorphic to
     the functor that takes a 2-cell \<open>\<mu>\<close> of \<open>C\<close> to \<open>\<sigma>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)\<close>
     and symmetrically for \<open>G\<close>.  Here we just establish the naturality property and
     and that each 1-cell \<open>\<guillemotleft>g : a \<rightarrow>\<^sub>C a'\<guillemotright>\<close> is isomorphic to \<open>\<tau>\<^sub>0 a' \<star>\<^sub>D (\<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a\<close>.
     We ultimately need these results to prove that a pseudofunctor extends to an
     equivalence of bicategories if and only if it is an equivalence pseudofunctor.
   \<close>
 
   context pseudonatural_equivalence
   begin
 
     interpretation \<sigma>: converse_pseudonatural_equivalence
                         V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
       ..
 
     abbreviation (input) \<sigma>\<^sub>0
     where "\<sigma>\<^sub>0 \<equiv> \<sigma>.map\<^sub>0"
 
     definition \<phi>
     where "\<phi> f \<equiv> \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C f), F f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                  (\<tau>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D
                  \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D
                  (G f \<star>\<^sub>D D.inv (\<sigma>.\<epsilon> (src\<^sub>C f))) \<cdot>\<^sub>D
                  \<r>\<^sub>D\<^sup>-\<^sup>1[G f]"
 
     lemma \<phi>_in_hom [intro]:
     assumes "C.ide f"
     shows "\<guillemotleft>\<phi> f : G f \<Rightarrow>\<^sub>D \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)\<guillemotright>"
       unfolding \<phi>_def
       using assms by (intro D.comp_in_homI' D.hseqI') auto
 
     lemma iso_\<phi>:
     assumes "C.ide f"
     shows "D.iso (\<phi> f)"
       unfolding \<phi>_def
       using assms iso_map\<^sub>1_ide
       by (intro D.isos_compose) auto
 
     definition \<psi>
     where "\<psi> f \<equiv> (\<sigma>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D
                   \<a>\<^sub>D[\<sigma>\<^sub>0 (trg\<^sub>C f), \<tau>\<^sub>0 (trg\<^sub>C f), F f] \<cdot>\<^sub>D
                   (\<sigma>.\<eta> (trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D
                   \<l>\<^sub>D\<^sup>-\<^sup>1[F f]"
 
     lemma \<psi>_in_hom [intro]:
     assumes "C.ide f"
     shows "\<guillemotleft>\<psi> f : F f \<Rightarrow>\<^sub>D \<sigma>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f)\<guillemotright>"
       unfolding \<psi>_def
       using assms iso_map\<^sub>1_ide
       by (intro D.comp_in_homI' D.hseqI') auto
 
     lemma iso_\<psi>:
     assumes "C.ide f"
     shows "D.iso (\<psi> f)"
       unfolding \<psi>_def
       using assms iso_map\<^sub>1_ide
       by (intro D.isos_compose) auto
 
     lemma \<psi>_naturality:
     assumes "C.arr \<mu>"
     shows "(\<sigma>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D \<psi> (C.dom \<mu>) = \<psi> (C.cod \<mu>) \<cdot>\<^sub>D F \<mu>"
     and "D.inv (\<psi> (C.cod \<mu>)) \<cdot>\<^sub>D (\<sigma>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D \<psi> (C.dom \<mu>) = F \<mu>"
     proof -
       let ?a = "src\<^sub>C \<mu>" and ?a' = "trg\<^sub>C \<mu>"
       let ?f = "C.dom \<mu>" and ?f' = "C.cod \<mu>"
       have "(\<sigma>\<^sub>0 ?a' \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<psi> (C.dom \<mu>)
               = ((\<sigma>\<^sub>0 ?a' \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D
                 (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f))) \<cdot>\<^sub>D
                 \<a>\<^sub>D[\<sigma>\<^sub>0 ?a', \<tau>\<^sub>0 ?a', F ?f] \<cdot>\<^sub>D
                 (\<sigma>.\<eta> ?a' \<star>\<^sub>D F ?f) \<cdot>\<^sub>D
                 \<l>\<^sub>D\<^sup>-\<^sup>1[F ?f]"
         unfolding \<psi>_def       
         using assms D.comp_assoc by simp
       also have "... = (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f')) \<cdot>\<^sub>D
                        ((\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D
                        \<a>\<^sub>D[\<sigma>\<^sub>0 ?a', \<tau>\<^sub>0 ?a', F ?f]) \<cdot>\<^sub>D
                        (\<sigma>.\<eta> ?a' \<star>\<^sub>D F ?f) \<cdot>\<^sub>D
                        \<l>\<^sub>D\<^sup>-\<^sup>1[F ?f]"            
       proof -
         have "(\<sigma>\<^sub>0 ?a' \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f))
                 = \<sigma>\<^sub>0 ?a' \<star>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)"
           using assms D.whisker_left iso_map\<^sub>1_ide by simp
         also have "... = \<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f') \<cdot>\<^sub>D (\<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>)"
           using assms naturality iso_map\<^sub>1_ide
                 D.invert_opposite_sides_of_square
                    [of "\<tau>\<^sub>1 ?f'" "G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a" "\<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>" "\<tau>\<^sub>1 ?f"]
           by simp
         also have "... = (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f')) \<cdot>\<^sub>D (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>)"
           using assms D.whisker_left iso_map\<^sub>1_ide by simp
         finally have "(\<sigma>\<^sub>0 ?a' \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f))
                         = (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f')) \<cdot>\<^sub>D (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>)"
           by blast
         thus ?thesis
           using D.comp_assoc by simp
       qed
       also have "... = (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f')) \<cdot>\<^sub>D
                        \<a>\<^sub>D[\<sigma>\<^sub>0 ?a', \<tau>\<^sub>0 ?a', F ?f'] \<cdot>\<^sub>D
                        (((\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a') \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D
                        (\<sigma>.\<eta> ?a' \<star>\<^sub>D F ?f)) \<cdot>\<^sub>D
                        \<l>\<^sub>D\<^sup>-\<^sup>1[F ?f]"            
       proof -
         have "(\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<a>\<^sub>D[\<sigma>\<^sub>0 ?a', \<tau>\<^sub>0 ?a', F ?f]
                 = \<a>\<^sub>D[\<sigma>\<^sub>0 ?a', \<tau>\<^sub>0 ?a', F ?f'] \<cdot>\<^sub>D ((\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a') \<star>\<^sub>D F \<mu>)"
           using assms D.assoc_naturality [of "\<sigma>\<^sub>0 ?a'" "\<tau>\<^sub>0 ?a'" "F \<mu>"] by simp
         thus ?thesis
           using D.comp_assoc by simp
       qed
       also have "... = (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D D.inv (\<tau>\<^sub>1 ?f')) \<cdot>\<^sub>D
                        \<a>\<^sub>D[\<sigma>\<^sub>0 ?a', \<tau>\<^sub>0 ?a', F ?f'] \<cdot>\<^sub>D
                        (\<sigma>.\<eta> ?a' \<star>\<^sub>D F ?f') \<cdot>\<^sub>D
                        (F.map\<^sub>0 ?a' \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D
                        \<l>\<^sub>D\<^sup>-\<^sup>1[F ?f]"            
       proof -
         have "((\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a') \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D (\<sigma>.\<eta> ?a' \<star>\<^sub>D F ?f)
                 = (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a') \<cdot>\<^sub>D \<sigma>.\<eta> ?a' \<star>\<^sub>D F \<mu> \<cdot>\<^sub>D F ?f"
           using assms D.interchange [of "\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a'" "\<sigma>.\<eta> ?a'" "F \<mu>" "F ?f"]
           by simp
         also have "... = \<sigma>.\<eta> ?a' \<cdot>\<^sub>D F.map\<^sub>0 ?a' \<star>\<^sub>D F ?f' \<cdot>\<^sub>D F \<mu>"
           using assms D.comp_arr_dom D.comp_cod_arr by simp
         also have "... = (\<sigma>.\<eta> ?a' \<star>\<^sub>D F ?f') \<cdot>\<^sub>D (F.map\<^sub>0 ?a' \<star>\<^sub>D F \<mu>)"
           using assms D.interchange [of "\<sigma>.\<eta> ?a'" "F.map\<^sub>0 ?a'" "F ?f'" "F \<mu>"]
                 \<sigma>.unit_in_hom [of ?a']
           by fastforce
         finally have "((\<sigma>\<^sub>0 ?a' \<star>\<^sub>D \<tau>\<^sub>0 ?a') \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D (\<sigma>.\<eta> ?a' \<star>\<^sub>D F ?f)
                         = (\<sigma>.\<eta> ?a' \<star>\<^sub>D F ?f') \<cdot>\<^sub>D (F.map\<^sub>0 ?a' \<star>\<^sub>D F \<mu>)"
           by blast
         thus ?thesis
           using D.comp_assoc by simp
       qed
       also have "... = \<psi> ?f' \<cdot>\<^sub>D F \<mu>"            
       proof -
         have "(F.map\<^sub>0 ?a' \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F ?f] = \<l>\<^sub>D\<^sup>-\<^sup>1[F ?f'] \<cdot>\<^sub>D F \<mu>"
           using assms D.lunit'_naturality [of "F \<mu>"] by simp
         thus ?thesis
         unfolding \<psi>_def       
           using assms D.comp_assoc by simp
       qed
       finally show "(\<sigma>\<^sub>0 ?a' \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<psi> (C.dom \<mu>) = \<psi> ?f' \<cdot>\<^sub>D F \<mu>"
         by blast
       thus "D.inv (\<psi> ?f') \<cdot>\<^sub>D (\<sigma>\<^sub>0 ?a' \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<psi> (C.dom \<mu>) = F \<mu>"
         using assms \<psi>_in_hom iso_\<psi>
         by (metis C.ide_cod D.in_homE D.invert_side_of_triangle(1) D.seqI
             F.preserves_arr F.preserves_cod)
     qed
 
     lemma isomorphic_expansion:
     assumes "C.obj a" and "C.obj a'" and "\<guillemotleft>g : G.map\<^sub>0 a \<rightarrow>\<^sub>D G.map\<^sub>0 a'\<guillemotright>" and "D.ide g"
     shows "D.isomorphic (\<tau>\<^sub>0 a' \<star>\<^sub>D (\<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a) g"
     proof -
       let ?g' = "\<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a"
       have g': "\<guillemotleft>?g' : F.map\<^sub>0 a \<rightarrow>\<^sub>D F.map\<^sub>0 a'\<guillemotright>"
         using assms ide_map\<^sub>0_obj \<sigma>.map\<^sub>0_simps(3) C.obj_simps
         by (intro D.in_hhomI D.hseqI') auto
       have ide_g': "D.ide ?g'"
         using assms g' \<sigma>.ide_map\<^sub>0_obj ide_map\<^sub>0_obj by blast
       let ?\<psi> = "(\<sigma>\<^sub>0 a' \<star>\<^sub>D \<r>\<^sub>D[g]) \<cdot>\<^sub>D (\<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<sigma>.\<epsilon> a) \<cdot>\<^sub>D
                 (\<sigma>\<^sub>0 a' \<star>\<^sub>D \<a>\<^sub>D[g, \<tau>\<^sub>0 a, \<sigma>\<^sub>0 a]) \<cdot>\<^sub>D \<a>\<^sub>D[\<sigma>\<^sub>0 a', g \<star>\<^sub>D \<tau>\<^sub>0 a, \<sigma>\<^sub>0 a]"
       have \<psi>: "\<guillemotleft>?\<psi> : ?g' \<star>\<^sub>D \<sigma>\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>\<^sub>0 a' \<star>\<^sub>D g\<guillemotright>"
       proof (intro D.comp_in_homI)
         show "\<guillemotleft>\<a>\<^sub>D[\<sigma>\<^sub>0 a', g \<star>\<^sub>D \<tau>\<^sub>0 a, \<sigma>\<^sub>0 a] :
                  (\<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>\<^sub>0 a' \<star>\<^sub>D (g \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a\<guillemotright>"
           using assms g' by fastforce
         show "\<guillemotleft>\<sigma>\<^sub>0 a' \<star>\<^sub>D \<a>\<^sub>D[g, \<tau>\<^sub>0 a, \<sigma>\<^sub>0 a] :
                  \<sigma>\<^sub>0 a' \<star>\<^sub>D (g \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<sigma>\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a\<guillemotright>"
           using assms g' by fastforce
         show "\<guillemotleft>\<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<sigma>.\<epsilon> a :
                  \<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<sigma>\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D G.map\<^sub>0 a\<guillemotright>"
           using assms C.obj_simps
           by (intro D.hcomp_in_vhom) auto
         show "\<guillemotleft>\<sigma>\<^sub>0 a' \<star>\<^sub>D \<r>\<^sub>D[g] : \<sigma>\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D G.map\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>\<^sub>0 a' \<star>\<^sub>D g\<guillemotright>"
           using assms
           apply (intro D.hcomp_in_vhom)
             apply auto
           by fastforce
       qed
       have iso_\<psi>: "D.iso ?\<psi>"
         using assms g' \<psi> ide_g' ide_map\<^sub>0_obj C.obj_simps
         by (intro D.isos_compose) auto
       let ?\<phi> = "\<l>\<^sub>D[g] \<cdot>\<^sub>D (\<sigma>.\<epsilon> a' \<star>\<^sub>D g) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a', \<sigma>\<^sub>0 a', g] \<cdot>\<^sub>D (\<tau>\<^sub>0 a' \<star>\<^sub>D ?\<psi>)"
       have \<phi>: "\<guillemotleft>?\<phi> : \<tau>\<^sub>0 a' \<star>\<^sub>D ?g' \<star>\<^sub>D \<sigma>\<^sub>0 a \<Rightarrow>\<^sub>D g\<guillemotright>"
       proof (intro D.comp_in_homI)
         show "\<guillemotleft>\<tau>\<^sub>0 a' \<star>\<^sub>D ?\<psi> : \<tau>\<^sub>0 a' \<star>\<^sub>D ?g' \<star>\<^sub>D \<sigma>\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0 a' \<star>\<^sub>D \<sigma>\<^sub>0 a' \<star>\<^sub>D g\<guillemotright>"
           using assms g' \<psi> C.obj_simps
           by (intro D.hcomp_in_vhom) auto
         show "\<guillemotleft>\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a', \<sigma>\<^sub>0 a', g] : \<tau>\<^sub>0 a' \<star>\<^sub>D \<sigma>\<^sub>0 a' \<star>\<^sub>D g \<Rightarrow>\<^sub>D (\<tau>\<^sub>0 a' \<star>\<^sub>D \<sigma>\<^sub>0 a') \<star>\<^sub>D g\<guillemotright>"
         proof -
           have "src\<^sub>D (\<tau>\<^sub>0 a') = trg\<^sub>D (\<sigma>\<^sub>0 a')"
             using assms by auto
           moreover have "src\<^sub>D (\<sigma>\<^sub>0 a') = trg\<^sub>D g"
             using assms by auto
           ultimately show ?thesis
             using assms ide_map\<^sub>0_obj \<sigma>.ide_map\<^sub>0_obj by auto
         qed
         show "\<guillemotleft>\<sigma>.\<epsilon> a' \<star>\<^sub>D g : (\<tau>\<^sub>0 a' \<star>\<^sub>D \<sigma>\<^sub>0 a') \<star>\<^sub>D g \<Rightarrow>\<^sub>D G.map\<^sub>0 a' \<star>\<^sub>D g\<guillemotright>"
           using assms by fastforce
         show "\<guillemotleft>\<l>\<^sub>D[g] : G.map\<^sub>0 a' \<star>\<^sub>D g \<Rightarrow>\<^sub>D g\<guillemotright>"
           using assms by auto
       qed
       have iso_\<phi>: "D.iso ?\<phi>"
         using assms g' \<phi> \<psi> iso_\<psi> ide_map\<^sub>0_obj iso_map\<^sub>1_ide \<sigma>.ide_map\<^sub>0_obj
         apply (intro D.isos_compose)
               apply (meson D.arrI D.hseqE D.ide_is_iso D.iso_hcomp D.seqE)
              apply (metis D.hseqE D.in_hhomE D.iso_assoc' D.trg_hcomp F.map\<^sub>0_def map\<^sub>0_simps(2))
         by auto
       show ?thesis
         using \<phi> iso_\<phi> D.isomorphic_def by auto
     qed
 
   end
 
   text \<open>
     Here we show that pseudonatural equivalence respects equivalence pseudofunctors,
     in the sense that a pseudofunctor \<open>G\<close>, pseudonaturally equivalent to an
     equivalence pseudofunctor \<open>F\<close>, is itself an equivalence pseudofunctor.
   \<close>
 
   locale pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor =
     C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C +
     D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D +
     F: equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F +
     pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G +
     \<tau>: pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
   for V\<^sub>C :: "'c comp"                   (infixr "\<cdot>\<^sub>C" 55)
   and H\<^sub>C :: "'c comp"                   (infixr "\<star>\<^sub>C" 53)
   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"       ("\<a>\<^sub>C[_, _, _]")
   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"                   ("\<i>\<^sub>C[_]")
   and src\<^sub>C :: "'c \<Rightarrow> 'c"
   and trg\<^sub>C :: "'c \<Rightarrow> 'c"
   and V\<^sub>D :: "'d comp"                   (infixr "\<cdot>\<^sub>D" 55)
   and H\<^sub>D :: "'d comp"                   (infixr "\<star>\<^sub>D" 53)
   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"       ("\<a>\<^sub>D[_, _, _]")
   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"                   ("\<i>\<^sub>D[_]")
   and src\<^sub>D :: "'d \<Rightarrow> 'd"
   and trg\<^sub>D :: "'d \<Rightarrow> 'd"
   and F :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>F :: "'c * 'c \<Rightarrow> 'd"
   and G :: "'c \<Rightarrow> 'd"
   and \<Phi>\<^sub>G :: "'c * 'c \<Rightarrow> 'd"
   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"
   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd"
   begin
 
     interpretation \<sigma>': converse_pseudonatural_equivalence
                         V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
       ..
 
     sublocale G: equivalence_pseudofunctor
                    V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G
     proof
       show "\<And>\<mu> \<mu>'. \<lbrakk>C.par \<mu> \<mu>'; G \<mu> = G \<mu>'\<rbrakk> \<Longrightarrow> \<mu> = \<mu>'"
       proof -
         fix \<mu> \<mu>'
         assume par: "C.par \<mu> \<mu>'" and eq: "G \<mu> = G \<mu>'"
         let ?a = "src\<^sub>C \<mu>" and ?a' = "trg\<^sub>C \<mu>"
         let ?f = "C.dom \<mu>" and ?f' = "C.cod \<mu>"
         have "\<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu> = (\<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<tau>\<^sub>1 ?f \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)"
           using par \<tau>.ide_map\<^sub>0_obj \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_arr_inv'
                 \<tau>.map\<^sub>1_in_hom [of \<mu>]
           by (metis C.dom_trg C.ide_dom C.obj_trg C.trg.preserves_dom D.whisker_left
               F.is_natural_2 F.naturality F.preserves_arr \<tau>.map\<^sub>1_simps(5))
         also have "... = ((\<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<tau>\<^sub>1 ?f) \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)"
           using D.comp_assoc by simp
         also have "... = ((\<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>') \<cdot>\<^sub>D \<tau>\<^sub>1 ?f) \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)"
           using eq par \<tau>.naturality [of \<mu>] \<tau>.naturality [of \<mu>'] C.src_cod C.trg_cod by metis
         also have 1: "... = (\<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>') \<cdot>\<^sub>D \<tau>\<^sub>1 ?f \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 ?f)"
           using D.comp_assoc by blast
         also have "... = \<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>'"
           using par \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_arr_inv' \<tau>.map\<^sub>1_in_hom [of \<mu>']
           by (metis 1 C.ide_dom C.trg_cod D.hseqE D.hseqI' D.hseq_char' D.seqE
               F.preserves_arr F.preserves_trg calculation)
         finally have "\<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu> = \<tau>\<^sub>0 ?a' \<star>\<^sub>D F \<mu>'"
           by blast
         hence "F \<mu> = F \<mu>'"
           using par \<tau>.components_are_equivalences
                 D.equivalence_cancel_left [of "\<tau>\<^sub>0 ?a'" "F \<mu>" "F \<mu>'"]
           by simp
         thus "\<mu> = \<mu>'"
           using par F.is_faithful by blast
       qed
       show "\<And>b. D.obj b \<Longrightarrow> \<exists>a. C.obj a \<and> D.equivalent_objects (map\<^sub>0 a) b"
       proof -
         fix b
         assume b: "D.obj b"
         obtain a where a: "C.obj a \<and> D.equivalent_objects (F.map\<^sub>0 a) b"
           using b F.biessentially_surjective_on_objects by blast
         have "D.equivalent_objects (F.map\<^sub>0 a) (map\<^sub>0 a)"
           using a \<tau>.components_are_equivalences D.equivalent_objects_def
           by (metis F.map\<^sub>0_def map\<^sub>0_def \<tau>.map\<^sub>0_in_hhom)
         hence "D.equivalent_objects (map\<^sub>0 a) b"
           using a D.equivalent_objects_symmetric D.equivalent_objects_transitive by blast
         thus "\<exists>a. C.obj a \<and> D.equivalent_objects (map\<^sub>0 a) b"
           using a by auto
       qed
       show "\<And>a a' g. \<lbrakk>C.obj a; C.obj a'; \<guillemotleft>g : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a'\<guillemotright>; D.ide g\<rbrakk>
                        \<Longrightarrow> \<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C a'\<guillemotright> \<and> C.ide f \<and> D.isomorphic (G f) g"
       proof -
         fix a a' g
         assume a: "C.obj a" and a': "C.obj a'"
         assume g: "\<guillemotleft>g : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a'\<guillemotright>" and ide_g: "D.ide g"
         interpret \<sigma>: converse_pseudonatural_equivalence
                         V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
           ..
         interpret \<sigma>: pseudonatural_equivalence
                        V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
                        G \<Phi>\<^sub>G F \<Phi>\<^sub>F \<sigma>'.map\<^sub>0 \<sigma>'.map\<^sub>1
           using \<sigma>'.is_pseudonatural_equivalence by simp
         let ?g' = "\<sigma>'.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a"
         have g': "\<guillemotleft>?g' : F.map\<^sub>0 a \<rightarrow>\<^sub>D F.map\<^sub>0 a'\<guillemotright>"
           using a a' g \<tau>.ide_map\<^sub>0_obj \<sigma>'.map\<^sub>0_simps(3) C.obj_simps
           by (intro D.in_hhomI D.hseqI') auto
         have ide_g': "D.ide ?g'"
           using a a' g g' ide_g
           by (meson D.hcomp_in_hhomE D.hseqE D.ide_hcomp \<sigma>'.ide_map\<^sub>0_obj \<tau>.ide_map\<^sub>0_obj)
         obtain f where f: "\<guillemotleft>f : a \<rightarrow>\<^sub>C a'\<guillemotright> \<and> C.ide f \<and> D.isomorphic (F f) ?g'"
           using a a' g g' ide_g' F.locally_essentially_surjective [of a a' ?g'] by auto
         obtain \<xi> where \<xi>: "\<guillemotleft>\<xi> : F f \<Rightarrow>\<^sub>D ?g'\<guillemotright> \<and> D.iso \<xi>"
           using f D.isomorphic_def by blast
         have "D.isomorphic (G f) g"
         proof -
           have "D.isomorphic (G f) (\<tau>\<^sub>0 a' \<star>\<^sub>D F f \<star>\<^sub>D \<sigma>'.map\<^sub>0 a)"
             using a a' f \<tau>.iso_\<phi> D.isomorphic_def by blast
           also have "D.isomorphic ... (\<tau>\<^sub>0 a' \<star>\<^sub>D ?g' \<star>\<^sub>D \<sigma>'.map\<^sub>0 a)"
             using a a' g' \<xi> D.isomorphic_def
             by (metis D.hcomp_ide_isomorphic D.hcomp_isomorphic_ide D.hseqE D.ideD(1)
                 D.isomorphic_implies_ide(2) \<sigma>'.ide_map\<^sub>0_obj calculation \<tau>.ide_map\<^sub>0_obj)
           also have "D.isomorphic ... g"
             using a a' g ide_g \<tau>.isomorphic_expansion by simp
           finally show ?thesis by blast
         qed
         thus "\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C a'\<guillemotright> \<and> C.ide f \<and> D.isomorphic (G f) g"
           using f by auto
       qed
       show "\<And>f f' \<nu>. \<lbrakk>C.ide f; C.ide f'; src\<^sub>C f = src\<^sub>C f'; trg\<^sub>C f = trg\<^sub>C f'; \<guillemotleft>\<nu> : G f \<Rightarrow>\<^sub>D G f'\<guillemotright>\<rbrakk>
                          \<Longrightarrow> \<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C f'\<guillemotright> \<and> G \<mu> = \<nu>"
       proof -
         fix f f' \<nu>
         assume f: "C.ide f" and f': "C.ide f'"
         and eq_src: "src\<^sub>C f = src\<^sub>C f'" and eq_trg: "trg\<^sub>C f = trg\<^sub>C f'"
         and \<nu>: "\<guillemotleft>\<nu> : G f \<Rightarrow>\<^sub>D G f'\<guillemotright>"
         let ?a = "src\<^sub>C f" and ?a' = "trg\<^sub>C f"
         let ?\<nu>' = "D.inv (\<tau>.\<psi> f') \<cdot>\<^sub>D (\<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<tau>.\<psi> f"
         have \<nu>': "\<guillemotleft>?\<nu>' : F f \<Rightarrow>\<^sub>D F f'\<guillemotright>"
         proof (intro D.comp_in_homI)
           show "\<guillemotleft>\<tau>.\<psi> f : F f \<Rightarrow>\<^sub>D \<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a\<guillemotright>"
             using f \<tau>.\<psi>_in_hom [of f] by blast
           show "\<guillemotleft>\<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a :
                    \<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 ?a \<Rightarrow>\<^sub>D \<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D G f' \<star>\<^sub>D \<tau>\<^sub>0 ?a\<guillemotright>"
             using f \<nu> by (intro D.hcomp_in_vhom) auto
           show "\<guillemotleft>D.inv (\<tau>.\<psi> f') : \<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D G f' \<star>\<^sub>D \<tau>\<^sub>0 ?a \<Rightarrow>\<^sub>D F f'\<guillemotright>"
             using f' \<tau>.iso_\<psi> eq_src eq_trg by auto
         qed
         obtain \<mu> where \<mu>: "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C f'\<guillemotright> \<and> F \<mu> = ?\<nu>'"
           using f f' eq_src eq_trg \<nu>' F.locally_full [of f f' ?\<nu>'] by blast
         have "G \<mu> = \<nu>"
         proof -
           have "D.inv (\<tau>.\<psi> f') \<cdot>\<^sub>D (\<sigma>'.map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D \<tau>.\<psi> f =
                 D.inv (\<tau>.\<psi> f') \<cdot>\<^sub>D (\<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<tau>.\<psi> f"
             using \<mu> \<tau>.\<psi>_naturality [of \<mu>] by auto
           hence "\<sigma>'.map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>) = \<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a"
             using f f' \<nu>' \<tau>.iso_\<psi>
                   D.iso_cancel_left
                     [of "D.inv (\<tau>.\<psi> f')" "(\<sigma>'.map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D \<tau>.\<psi> f"
                         "(\<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a) \<cdot>\<^sub>D \<tau>.\<psi> f"]
                   D.iso_cancel_right
                     [of "\<tau>.\<psi> f" "\<sigma>'.map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)" "\<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a"]
             by auto
           hence "\<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a = \<sigma>'.map\<^sub>0 ?a' \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a"
             using \<mu> by auto
           moreover have "D.par (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f)) (\<nu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f))"
             using f \<mu> \<nu> \<nu>'
             apply (intro conjI D.hseqI')
                    apply (auto simp add: D.vconn_implies_hpar(1))
              apply fastforce
             by (metis C.ideD(1) C.vconn_implies_hpar(1) D.hcomp_simps(4) D.hseqE D.hseqI'
                 D.seqE D.vconn_implies_hpar(1) preserves_arr preserves_cod preserves_src
                 weak_arrow_of_homs_axioms category.in_homE horizontal_homs_def
                 weak_arrow_of_homs_def)
           ultimately have "G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a = \<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a"
             using f f' \<mu> \<nu> \<tau>.ide_map\<^sub>0_obj \<sigma>'.ide_map\<^sub>0_obj
                   D.equivalence_cancel_left [of "\<sigma>'.map\<^sub>0 ?a'" "G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 ?a" "\<nu> \<star>\<^sub>D \<tau>\<^sub>0 ?a"]
                   \<sigma>'.components_are_equivalences
             by auto
           moreover have "D.par (G \<mu>) \<nu>"
             using f f' \<mu> \<nu> by fastforce
           ultimately show ?thesis
             using f f' \<mu> \<nu> \<tau>.ide_map\<^sub>0_obj
                   D.equivalence_cancel_right [of "\<tau>\<^sub>0 ?a" "G \<mu>" "\<nu>"] \<tau>.components_are_equivalences
             by auto
         qed
         thus "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C f'\<guillemotright> \<and> G \<mu> = \<nu>"
           using \<mu> by auto
       qed
     qed
 
     lemma is_equivalence_pseudofunctor:
     shows "equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"
       ..
 
   end
 
   lemma pseudonaturally_equivalent_respects_equivalence_pseudofunctor:
   assumes "pseudonaturally_equivalent
              V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G"
   and "equivalence_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
   shows "equivalence_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G"
   proof -
     obtain \<tau>\<^sub>0 \<tau>\<^sub>1 where \<tau>: "pseudonatural_equivalence
                             V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
       using assms pseudonaturally_equivalent_def by blast
     interpret F: equivalence_pseudofunctor
                    V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F
       using assms by simp
     interpret G: pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G
       using assms \<tau>
       by (simp add: pseudonatural_equivalence_def pseudonatural_transformation_def)
     interpret \<tau>: pseudonatural_equivalence
                    V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
       using \<tau> by simp
     interpret G: pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor
                    V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1
       ..
     show "equivalence_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G"
       using G.is_equivalence_pseudofunctor by simp
   qed
 
 end
diff --git a/thys/Category3/CartesianCategory.thy b/thys/Category3/CartesianCategory.thy
--- a/thys/Category3/CartesianCategory.thy
+++ b/thys/Category3/CartesianCategory.thy
@@ -1,2285 +1,2285 @@
 (*  Title:       CartesianCategory
     Author:      Eugene W. Stark <stark@cs.stonybrook.edu>, 2020
     Maintainer:  Eugene W. Stark <stark@cs.stonybrook.edu>
 *)
 
 chapter "Cartesian Category"
 
 text\<open>
   In this chapter, we explore the notion of a ``cartesian category'', which we define
   to be a category having binary products and a terminal object.
   We show that every cartesian category extends to an ``elementary cartesian category'',
   whose definition assumes that specific choices have been made for projections and
   terminal object.
   Conversely, the underlying category of an elementary cartesian category is a
   cartesian category.
   We also show that cartesian categories are the same thing as categories with
   finite products.
 \<close>
 
 theory CartesianCategory
-imports Category3.Limit Category3.SetCat
+imports Limit SetCat
 begin
 
   section "Category with Binary Products"
 
   subsection "Binary Product Diagrams"
 
   text \<open>
     The ``shape'' of a binary product diagram is a category having two distinct identity arrows
     and no non-identity arrows.
   \<close>
 
   locale binary_product_shape
   begin
 
     sublocale concrete_category \<open>UNIV :: bool set\<close> \<open>\<lambda>a b. if a = b then {()} else {}\<close>
                                 \<open>\<lambda>_. ()\<close> \<open>\<lambda>_ _ _ _ _. ()\<close>
       apply (unfold_locales, auto)
        apply (meson empty_iff)
       by (meson empty_iff)
 
     abbreviation comp
     where "comp \<equiv> COMP"
 
     abbreviation FF
     where "FF \<equiv> MkIde False"
 
     abbreviation TT
     where "TT \<equiv> MkIde True"
 
     lemma arr_char:
     shows "arr f \<longleftrightarrow> f = FF \<or> f = TT"
       using arr_char by (cases f, simp_all)
 
     lemma ide_char:
     shows "ide f \<longleftrightarrow> f = FF \<or> f = TT"
       using ide_char ide_MkIde by (cases f, auto)
 
     lemma is_discrete:
     shows "ide f \<longleftrightarrow> arr f"
       using arr_char ide_char by simp
 
     lemma dom_simp [simp]:
     assumes "arr f"
     shows "dom f = f"
       using assms is_discrete by simp
 
     lemma cod_simp [simp]:
     assumes "arr f"
     shows "cod f = f"
       using assms is_discrete by simp
 
     lemma seq_char:
     shows "seq f g \<longleftrightarrow> arr f \<and> f = g"
       by auto
 
     lemma comp_simp [simp]:
     assumes "seq f g"
     shows "comp f g = f"
       using assms seq_char by fastforce
 
   end
 
   locale binary_product_diagram =
     J: binary_product_shape +
     C: category C
   for C :: "'c comp"      (infixr "\<cdot>" 55)
   and a0 :: 'c
   and a1 :: 'c +
   assumes is_discrete: "C.ide a0 \<and> C.ide a1"
   begin
 
     notation J.comp      (infixr "\<cdot>\<^sub>J" 55)
 
     fun map
     where "map J.FF = a0"
         | "map J.TT = a1"
         | "map _ = C.null"
 
     sublocale diagram J.comp C map
     proof
       show "\<And>f. \<not> J.arr f \<Longrightarrow> map f = C.null"
         using J.arr_char map.elims by auto
       fix f
       assume f: "J.arr f"
       show "C.arr (map f)"
         using f J.arr_char is_discrete C.ideD(1) map.simps(1-2) by metis
       show "C.dom (map f) = map (J.dom f)"
         using f J.arr_char J.dom_char is_discrete by force
       show "C.cod (map f) = map (J.cod f)"
         using f J.arr_char J.cod_char is_discrete by force
       next
       fix f g
       assume fg: "J.seq g f"
       show "map (g \<cdot>\<^sub>J f) = map g \<cdot> map f"
         using fg J.arr_char J.seq_char J.null_char J.not_arr_null is_discrete
         by (metis (no_types, lifting) C.comp_ide_self J.comp_simp map.simps(1-2))
     qed
 
   end
 
   subsection "Category with Binary Products"
 
   text \<open>
     A \emph{binary product} in a category @{term C} is a limit of a binary product diagram
     in @{term C}.
   \<close>
 
   context binary_product_diagram
   begin
 
     definition mkCone
     where "mkCone p0 p1 \<equiv> \<lambda>j. if j = J.FF then p0 else if j = J.TT then p1 else C.null"
 
     abbreviation is_rendered_commutative_by
     where "is_rendered_commutative_by p0 p1 \<equiv>
            C.seq a0 p0 \<and> C.seq a1 p1 \<and> C.dom p0 = C.dom p1"
 
     abbreviation has_as_binary_product
     where "has_as_binary_product p0 p1 \<equiv> limit_cone (C.dom p0) (mkCone p0 p1)"
 
     lemma cone_mkCone:
     assumes "is_rendered_commutative_by p0 p1"
     shows "cone (C.dom p0) (mkCone p0 p1)"
     proof -
       interpret E: constant_functor J.comp C \<open>C.dom p0\<close>
         using assms by unfold_locales auto
       show "cone (C.dom p0) (mkCone p0 p1)"
         using assms mkCone_def J.arr_char E.map_simp is_discrete C.comp_ide_arr C.comp_arr_dom
         by unfold_locales auto
     qed
 
     lemma is_rendered_commutative_by_cone:
     assumes "cone a \<chi>"
     shows "is_rendered_commutative_by (\<chi> J.FF) (\<chi> J.TT)"
     proof -
       interpret \<chi>: cone J.comp C map a \<chi>
         using assms by auto
       show ?thesis
         using is_discrete by simp
     qed
 
     lemma mkCone_cone:
     assumes "cone a \<chi>"
     shows "mkCone (\<chi> J.FF) (\<chi> J.TT) = \<chi>"
     proof -
       interpret \<chi>: cone J.comp C map a \<chi>
         using assms by auto
       interpret mkCone_\<chi>: cone J.comp C map \<open>C.dom (\<chi> J.FF)\<close> \<open>mkCone (\<chi> J.FF) (\<chi> J.TT)\<close>
         using assms is_rendered_commutative_by_cone cone_mkCone by blast
       show ?thesis
         using mkCone_def \<chi>.is_extensional J.ide_char mkCone_def
               NaturalTransformation.eqI [of J.comp C]
               \<chi>.natural_transformation_axioms mkCone_\<chi>.natural_transformation_axioms
         by fastforce
     qed
 
   end
 
   locale binary_product_cone =
     J: binary_product_shape +
     C: category C +
     D: binary_product_diagram C f0 f1 +
     limit_cone J.comp C D.map \<open>C.dom p0\<close> \<open>D.mkCone p0 p1\<close>
   for C :: "'c comp"      (infixr "\<cdot>" 55)
   and f0 :: 'c
   and f1 :: 'c
   and p0 :: 'c
   and p1 :: 'c
   begin
 
     lemma renders_commutative:
     shows "D.is_rendered_commutative_by p0 p1"
       using cone_axioms D.is_rendered_commutative_by_cone D.mkCone_def \<Phi>.Ya.Cop_S.arr.simps(1)
       by (metis (no_types, lifting))  (* TODO: pretty opaque *)
 
     lemma is_universal':
     assumes "D.is_rendered_commutative_by p0' p1'"
     shows "\<exists>!h. \<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright> \<and> p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
     proof -
       have "D.cone (C.dom p0') (D.mkCone p0' p1')"
         using assms D.cone_mkCone by blast
       hence "\<exists>!h. \<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright> \<and>
                   D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1'"
         using is_universal by simp
       moreover have "\<And>h. \<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright> \<Longrightarrow>
                            D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1' \<longleftrightarrow>
                            p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
       proof -
         fix h
         assume h: "\<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright>"
         show "D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1' \<longleftrightarrow>
               p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
         proof
           assume 1: "D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1'"
           show "p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
           proof
             show "p0 \<cdot> h = p0'"
             proof -
               have "p0' = D.mkCone p0' p1' J.FF"
                 using D.mkCone_def J.arr_char by simp
               also have "... = D.cones_map h (D.mkCone p0 p1) J.FF"
                 using 1 by simp
               also have "... = p0 \<cdot> h"
                 using h D.mkCone_def J.arr_char cone_\<chi> by auto
               finally show ?thesis by auto
             qed
             show "p1 \<cdot> h = p1'"
             proof -
               have "p1' = D.mkCone p0' p1' J.TT"
                 using D.mkCone_def J.arr_char by simp
               also have "... = D.cones_map h (D.mkCone p0 p1) J.TT"
                 using 1 by simp
               also have "... = p1 \<cdot> h"
                 using h D.mkCone_def J.arr_char cone_\<chi> by auto
               finally show ?thesis by auto
             qed
           qed
           next
           assume 1: "p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
           show "D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1'"
             using h 1 cone_\<chi> D.mkCone_def by auto
         qed
       qed
       ultimately show ?thesis by blast
     qed
 
     lemma induced_arrowI':
     assumes "D.is_rendered_commutative_by p0' p1'"
     shows "\<guillemotleft>induced_arrow (C.dom p0') (D.mkCone p0' p1') : C.dom p0' \<rightarrow> C.dom p0\<guillemotright>"
     and "p0 \<cdot> induced_arrow (C.dom p0') (D.mkCone p0' p1') = p0'"
     and "p1 \<cdot> induced_arrow (C.dom p1') (D.mkCone p0' p1') = p1'"
     proof -
       interpret A': constant_functor J.comp C \<open>C.dom p0'\<close>
         using assms by (unfold_locales, auto)
       have cone: "D.cone (C.dom p0') (D.mkCone p0' p1')"
         using assms D.cone_mkCone [of p0' p1'] by blast
       show 0: "p0 \<cdot> induced_arrow (C.dom p0') (D.mkCone p0' p1') = p0'"
       proof -
         have "p0 \<cdot> induced_arrow (C.dom p0') (D.mkCone p0' p1') =
                 D.cones_map (induced_arrow (C.dom p0') (D.mkCone p0' p1'))
                             (D.mkCone p0 p1) J.FF"
           using cone induced_arrowI(1) D.mkCone_def J.arr_char cone_\<chi> by force
         also have "... = p0'"
         proof -
           have "D.cones_map (induced_arrow (C.dom p0') (D.mkCone p0' p1'))
                             (D.mkCone p0 p1) =
                 D.mkCone p0' p1'"
             using cone induced_arrowI by blast
           thus ?thesis
             using J.arr_char D.mkCone_def by simp
         qed
         finally show ?thesis by auto
       qed
       show "p1 \<cdot> induced_arrow (C.dom p1') (D.mkCone p0' p1') = p1'"
       proof -
         have "p1 \<cdot> induced_arrow (C.dom p1') (D.mkCone p0' p1') =
                 D.cones_map (induced_arrow (C.dom p0') (D.mkCone p0' p1'))
                             (D.mkCone p0 p1) J.TT"
           using assms cone induced_arrowI(1) D.mkCone_def J.arr_char cone_\<chi> by fastforce
         also have "... = p1'"
         proof -
           have "D.cones_map (induced_arrow (C.dom p0') (D.mkCone p0' p1'))
                             (D.mkCone p0 p1) =
                 D.mkCone p0' p1'"
             using cone induced_arrowI by blast
           thus ?thesis
             using J.arr_char D.mkCone_def by simp
         qed
         finally show ?thesis by auto
       qed
       show "\<guillemotleft>induced_arrow (C.dom p0') (D.mkCone p0' p1') : C.dom p0' \<rightarrow> C.dom p0\<guillemotright>"
         using 0 cone induced_arrowI by simp
     qed
 
   end
 
   context category
   begin
 
     definition has_as_binary_product
     where "has_as_binary_product a0 a1 p0 p1 \<equiv>
            ide a0 \<and> ide a1 \<and> binary_product_diagram.has_as_binary_product C a0 a1 p0 p1"
 
     definition has_binary_products
     where "has_binary_products =
            (\<forall>a0 a1. ide a0 \<and> ide a1 \<longrightarrow> (\<exists>p0 p1. has_as_binary_product a0 a1 p0 p1))"
 
   end
 
   locale category_with_binary_products =
     category +
   assumes has_binary_products: has_binary_products
 
   subsection "Elementary Category with Binary Products"
 
   text \<open>
     An \emph{elementary category with binary products} is a category equipped with a specific
     way of mapping each pair of objects \<open>a\<close> and \<open>b\<close> to a pair of arrows \<open>\<pp>\<^sub>1[a, b]\<close> and \<open>\<pp>\<^sub>0[a, b]\<close>
     that comprise a universal span.  It is useful to assume that the mappings that produce
     \<open>\<pp>\<^sub>1[a, b]\<close> and \<open>\<pp>\<^sub>0[a, b]\<close> from \<open>a\<close> and \<open>b\<close> are extensional; that is, if either \<open>a\<close> or \<open>b\<close>
     is not an identity, then \<open>\<pp>\<^sub>1[a, b]\<close> and \<open>\<pp>\<^sub>0[a, b]\<close> are \<open>null\<close>.
   \<close>
 
   locale elementary_category_with_binary_products =
     category C
   for C :: "'a comp"                             (infixr "\<cdot>" 55)
   and pr0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"                    ("\<pp>\<^sub>0[_, _]")
   and pr1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"                    ("\<pp>\<^sub>1[_, _]") +
   assumes pr0_ext: "\<not> (ide a \<and> ide b) \<Longrightarrow> \<pp>\<^sub>0[a, b] = null"
   and pr1_ext: "\<not> (ide a \<and> ide b) \<Longrightarrow> \<pp>\<^sub>1[a, b] = null"
   and span_pr: "\<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> span \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]"
   and cod_pr0: "\<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> cod \<pp>\<^sub>0[a, b] = b"
   and cod_pr1: "\<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> cod \<pp>\<^sub>1[a, b] = a"
   and universal: "span f g \<Longrightarrow> \<exists>!l. \<pp>\<^sub>1[cod f, cod g] \<cdot> l = f \<and> \<pp>\<^sub>0[cod f, cod g] \<cdot> l = g"
   begin
 
     lemma pr0_in_hom':
     assumes "ide a" and "ide b"
     shows "\<guillemotleft>\<pp>\<^sub>0[a, b] : dom \<pp>\<^sub>0[a, b] \<rightarrow> b\<guillemotright>"
       using assms span_pr cod_pr0 by auto
 
     lemma pr1_in_hom':
     assumes "ide a" and "ide b"
     shows "\<guillemotleft>\<pp>\<^sub>1[a, b] : dom \<pp>\<^sub>0[a, b] \<rightarrow> a\<guillemotright>"
       using assms span_pr cod_pr1 by auto
 
     text \<open>
       We introduce a notation for tupling, which denotes the arrow into a product that
       is induced by a span.
     \<close>
 
     definition tuple         ("\<langle>_, _\<rangle>")
     where "\<langle>f, g\<rangle> \<equiv> if span f g then
                       THE l. \<pp>\<^sub>1[cod f, cod g] \<cdot> l = f \<and> \<pp>\<^sub>0[cod f, cod g] \<cdot> l = g
                     else null"
 
     text \<open>
       The following defines product of arrows (not just of objects).  It will take a little
       while before we can prove that it is functorial, but for right now it is nice to have
       it as a notation for the apex of a product cone.  We have to go through some slightly
       unnatural contortions in the development here, though, to avoid having to introduce a
       separate preliminary notation just for the product of objects.
     \<close>
     (* TODO: I want to use \<times> but it has already been commandeered for product types. *)
     definition prod         (infixr "\<otimes>" 51)
     where "f \<otimes> g \<equiv> \<langle>f \<cdot> \<pp>\<^sub>1[dom f, dom g], g \<cdot> \<pp>\<^sub>0[dom f, dom g]\<rangle>"
 
     lemma seq_pr_tuple:
     assumes "span f g"
     shows "seq \<pp>\<^sub>0[cod f, cod g] \<langle>f, g\<rangle>"
     proof -
       have "\<pp>\<^sub>0[cod f, cod g] \<cdot> \<langle>f, g\<rangle> = g"
         unfolding tuple_def
         using assms universal theI [of "\<lambda>l. \<pp>\<^sub>1[cod f, cod g] \<cdot> l = f \<and> \<pp>\<^sub>0[cod f, cod g] \<cdot> l = g"]
         by simp meson
       thus ?thesis
         using assms by simp
     qed
 
     lemma tuple_pr_arr:
     assumes "ide a" and "ide b" and "seq \<pp>\<^sub>0[a, b] h"
     shows "\<langle>\<pp>\<^sub>1[a, b] \<cdot> h, \<pp>\<^sub>0[a, b] \<cdot> h\<rangle> = h"
       unfolding tuple_def
       using assms span_pr cod_pr0 cod_pr1 universal [of "\<pp>\<^sub>1[a, b] \<cdot> h" "\<pp>\<^sub>0[a, b] \<cdot> h"]
             theI_unique [of "\<lambda>l. \<pp>\<^sub>1[a, b] \<cdot> l = \<pp>\<^sub>1[a, b] \<cdot> h \<and> \<pp>\<^sub>0[a, b] \<cdot> l = \<pp>\<^sub>0[a, b] \<cdot> h" h]
       by auto
 
     lemma pr_tuple [simp]:
     assumes "span f g" and "cod f = a" and "cod g = b"
     shows "\<pp>\<^sub>1[a, b] \<cdot> \<langle>f, g\<rangle> = f" and "\<pp>\<^sub>0[a, b] \<cdot> \<langle>f, g\<rangle> = g"
     proof -
       have 1: "\<pp>\<^sub>1[a, b] \<cdot> \<langle>f, g\<rangle> = f \<and> \<pp>\<^sub>0[a, b] \<cdot> \<langle>f, g\<rangle> = g"
         unfolding tuple_def
         using assms universal theI [of "\<lambda>l. \<pp>\<^sub>1[a, b] \<cdot> l = f \<and> \<pp>\<^sub>0[a, b] \<cdot> l = g"]
         by simp meson
       show "\<pp>\<^sub>1[a, b] \<cdot> \<langle>f, g\<rangle> = f" using 1 by simp
       show "\<pp>\<^sub>0[a, b] \<cdot> \<langle>f, g\<rangle> = g" using 1 by simp
     qed
 
     lemma cod_tuple:
     assumes "span f g"
     shows "cod \<langle>f, g\<rangle> = cod f \<otimes> cod g"
     proof -
       have "cod f \<otimes> cod g = \<langle>\<pp>\<^sub>1[cod f, cod g], \<pp>\<^sub>0[cod f, cod g]\<rangle>"
         unfolding prod_def
         using assms comp_cod_arr span_pr cod_pr0 cod_pr1 by simp
       also have "... = \<langle>\<pp>\<^sub>1[cod f, cod g] \<cdot> dom \<pp>\<^sub>0[cod f, cod g],
                         \<pp>\<^sub>0[cod f, cod g] \<cdot> dom \<pp>\<^sub>0[cod f, cod g]\<rangle>"
         using assms span_pr comp_arr_dom by simp
       also have "... = dom \<pp>\<^sub>0[cod f, cod g]"
         using assms tuple_pr_arr span_pr by simp
       also have "... = cod \<langle>f, g\<rangle>"
         using assms seq_pr_tuple by blast
       finally show ?thesis by simp
     qed
 
     lemma tuple_in_hom [intro]:
     assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" and "\<guillemotleft>g : a \<rightarrow> c\<guillemotright>"
     shows "\<guillemotleft>\<langle>f, g\<rangle> : a \<rightarrow> b \<otimes> c\<guillemotright>"
       using assms pr_tuple dom_comp cod_tuple
       apply (elim in_homE, intro in_homI)
         apply (metis seqE)
       by metis+
 
     lemma tuple_in_hom' [simp]:
     assumes "arr f" and "dom f = a" and "cod f = b"
     and "arr g" and "dom g = a" and "cod g = c"
     shows "\<guillemotleft>\<langle>f, g\<rangle> : a \<rightarrow> b \<otimes> c\<guillemotright>"
       using assms by auto
 
     lemma tuple_ext:
     assumes "\<not> span f g"
     shows "\<langle>f, g\<rangle> = null"
       unfolding tuple_def
       by (simp add: assms)
 
     lemma tuple_simps [simp]:
     assumes "span f g"
     shows "arr \<langle>f, g\<rangle>" and "dom \<langle>f, g\<rangle> = dom f" and "cod \<langle>f, g\<rangle> = cod f \<otimes> cod g"
     proof -
       show "arr \<langle>f, g\<rangle>"
         using assms tuple_in_hom by blast
       show "dom \<langle>f, g\<rangle> = dom f"
         using assms tuple_in_hom
         by (metis dom_comp pr_tuple(1))
       show "cod \<langle>f, g\<rangle> = cod f \<otimes> cod g"
         using assms cod_tuple by auto
     qed
 
     lemma tuple_pr [simp]:
     assumes "ide a" and "ide b"
     shows "\<langle>\<pp>\<^sub>1[a, b], \<pp>\<^sub>0[a, b]\<rangle> = a \<otimes> b"
     proof -
       have 1: "dom \<pp>\<^sub>0[a, b] = a \<otimes> b"
         using assms seq_pr_tuple cod_tuple [of "\<pp>\<^sub>1[a, b]" "\<pp>\<^sub>0[a, b]"] span_pr
               pr0_in_hom' pr1_in_hom'
         by (metis cod_pr0 cod_pr1 seqE)
       hence "\<langle>\<pp>\<^sub>1[a, b], \<pp>\<^sub>0[a, b]\<rangle> = \<langle>\<pp>\<^sub>1[a, b] \<cdot> (a \<otimes> b), \<pp>\<^sub>0[a, b] \<cdot> (a \<otimes> b)\<rangle>"
         using assms pr0_in_hom' pr1_in_hom' comp_arr_dom span_pr by simp
       thus ?thesis
         using assms 1 tuple_pr_arr span_pr
         by (metis comp_arr_dom)
     qed
 
     lemma pr_in_hom [intro, simp]:
     assumes "ide a" and "ide b"
     shows "\<guillemotleft>\<pp>\<^sub>0[a, b] : a \<otimes> b \<rightarrow> b\<guillemotright>" and "\<guillemotleft>\<pp>\<^sub>1[a, b] : a \<otimes> b \<rightarrow> a\<guillemotright>"
     proof -
       show 0: "\<guillemotleft>\<pp>\<^sub>0[a, b] : a \<otimes> b \<rightarrow> b\<guillemotright>"
         using assms pr0_in_hom' seq_pr_tuple [of "\<pp>\<^sub>1[a, b]" "\<pp>\<^sub>0[a, b]"]
               cod_tuple [of "\<pp>\<^sub>1[a, b]" "\<pp>\<^sub>0[a, b]"] span_pr cod_pr0 cod_pr1
         by (intro in_homI, auto)
       show "\<guillemotleft>\<pp>\<^sub>1[a, b] : a \<otimes> b \<rightarrow> a\<guillemotright>"
         using assms 0 span_pr pr1_in_hom' by fastforce
     qed
 
     lemma pr_simps [simp]:
     assumes "ide a" and "ide b"
     shows "arr \<pp>\<^sub>0[a, b]" and "dom \<pp>\<^sub>0[a, b] = a \<otimes> b" and "cod \<pp>\<^sub>0[a, b] = b"
     and "arr \<pp>\<^sub>1[a, b]" and "dom \<pp>\<^sub>1[a, b] = a \<otimes> b" and "cod \<pp>\<^sub>1[a, b] = a"
       using assms pr_in_hom by blast+
 
     lemma arr_pr0_iff [iff]:
     shows "arr \<pp>\<^sub>0[a, b] \<longleftrightarrow> ide a \<and> ide b"
     proof
       show "ide a \<and> ide b \<Longrightarrow> arr \<pp>\<^sub>0[a, b]"
         using pr_in_hom by auto
       show "arr \<pp>\<^sub>0[a, b] \<Longrightarrow> ide a \<and> ide b"
         using pr0_ext not_arr_null by metis
     qed
 
     lemma arr_pr1_iff [iff]:
     shows "arr \<pp>\<^sub>1[a, b] \<longleftrightarrow> ide a \<and> ide b"
     proof
       show "ide a \<and> ide b \<Longrightarrow> arr \<pp>\<^sub>1[a, b]"
         using pr_in_hom by auto
       show "arr \<pp>\<^sub>1[a, b] \<Longrightarrow> ide a \<and> ide b"
         using pr1_ext not_arr_null by metis
     qed
 
     lemma pr_joint_monic:
     assumes "seq \<pp>\<^sub>0[a, b] h"
     and "\<pp>\<^sub>0[a, b] \<cdot> h = \<pp>\<^sub>0[a, b] \<cdot> h'" and "\<pp>\<^sub>1[a, b] \<cdot> h = \<pp>\<^sub>1[a, b] \<cdot> h'"
     shows "h = h'"
       using assms
       by (metis arr_pr0_iff seqE tuple_pr_arr)
 
     lemma comp_tuple_arr [simp]:
     assumes "span f g" and "arr h" and "dom f = cod h"
     shows "\<langle>f, g\<rangle> \<cdot> h = \<langle>f \<cdot> h, g \<cdot> h\<rangle>"
     proof (intro pr_joint_monic [where h = "\<langle>f, g\<rangle> \<cdot> h"])
       show "seq \<pp>\<^sub>0[cod f, cod g] (\<langle>f, g\<rangle> \<cdot> h)"
         using assms by fastforce
       show "\<pp>\<^sub>0[cod f, cod g] \<cdot> \<langle>f, g\<rangle> \<cdot> h = \<pp>\<^sub>0[cod f, cod g] \<cdot> \<langle>f \<cdot> h, g \<cdot> h\<rangle>"
       proof -
         have "\<pp>\<^sub>0[cod f, cod g] \<cdot> \<langle>f, g\<rangle> \<cdot> h = (\<pp>\<^sub>0[cod f, cod g] \<cdot> \<langle>f, g\<rangle>) \<cdot> h"
           using comp_assoc by simp
         thus ?thesis
           using assms by simp
       qed
       show "\<pp>\<^sub>1[cod f, cod g] \<cdot> \<langle>f, g\<rangle> \<cdot> h = \<pp>\<^sub>1[cod f, cod g] \<cdot> \<langle>f \<cdot> h, g \<cdot> h\<rangle>"
       proof -
         have "\<pp>\<^sub>1[cod f, cod g] \<cdot> \<langle>f, g\<rangle> \<cdot> h = (\<pp>\<^sub>1[cod f, cod g] \<cdot> \<langle>f, g\<rangle>) \<cdot> h"
           using comp_assoc by simp
         thus ?thesis
           using assms by simp
       qed
     qed
 
     lemma ide_prod [intro, simp]:
     assumes "ide a" and "ide b"
     shows "ide (a \<otimes> b)"
       using assms pr_simps ide_dom [of "\<pp>\<^sub>0[a, b]"] by simp
 
     lemma prod_in_hom [intro]:
     assumes "\<guillemotleft>f : a \<rightarrow> c\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow> d\<guillemotright>"
     shows "\<guillemotleft>f \<otimes> g : a \<otimes> b \<rightarrow> c \<otimes> d\<guillemotright>"
       using assms prod_def by fastforce
 
     lemma prod_in_hom' [simp]:
     assumes "arr f" and "dom f = a" and "cod f = c"
     and "arr g" and "dom g = b" and "cod g = d"
     shows "\<guillemotleft>f \<otimes> g : a \<otimes> b \<rightarrow> c \<otimes> d\<guillemotright>"
       using assms by blast
 
     lemma prod_simps [simp]:
     assumes "arr f0" and "arr f1"
     shows "arr (f0 \<otimes> f1)"
     and "dom (f0 \<otimes> f1) = dom f0 \<otimes> dom f1"
     and "cod (f0 \<otimes> f1) = cod f0 \<otimes> cod f1"
       using assms prod_in_hom by blast+
 
   end
 
   subsection "Agreement between the Definitions"
 
   text \<open>
     We now show that a category with binary products extends (by making a choice)
     to an elementary category with binary products, and that the underlying category
     of an elementary category with binary products is a category with binary products.
   \<close>
 
   context category_with_binary_products
   begin
 
     definition pr1
     where "pr1 a b \<equiv> if ide a \<and> ide b then
                         fst (SOME x. has_as_binary_product a b (fst x) (snd x))
                       else null"
 
     definition pr0
     where "pr0 a b \<equiv> if ide a \<and> ide b then
                         snd (SOME x. has_as_binary_product a b (fst x) (snd x))
                       else null"
 
     lemma pr_yields_binary_product:
     assumes "ide a" and "ide b"
     shows "has_as_binary_product a b (pr1 a b) (pr0 a b)"
     proof -
       have "\<exists>x. has_as_binary_product a b (fst x) (snd x)"
         using assms has_binary_products has_binary_products_def has_as_binary_product_def
         by simp
       thus ?thesis
         using assms has_binary_products has_binary_products_def pr0_def pr1_def
               someI_ex [of "\<lambda>x. has_as_binary_product a b (fst x) (snd x)"]
         by simp
     qed
 
     interpretation elementary_category_with_binary_products C pr0 pr1
     proof
       show "\<And>a b. \<not> (ide a \<and> ide b) \<Longrightarrow> pr0 a b = null"
         using pr0_def by auto
       show "\<And>a b. \<not> (ide a \<and> ide b) \<Longrightarrow> pr1 a b = null"
         using pr1_def by auto
       fix a b
       assume a: "ide a" and b: "ide b"
       interpret J: binary_product_shape .
       interpret D: binary_product_diagram C a b
         using a b by unfold_locales auto
       let ?\<chi> = "D.mkCone (pr1 a b) (pr0 a b)"
       interpret \<chi>: limit_cone J.comp C D.map \<open>dom (pr1 a b)\<close> ?\<chi>
         using a b pr_yields_binary_product
         by (simp add: has_as_binary_product_def)
       have 1: "pr1 a b = ?\<chi> J.FF \<and> pr0 a b = ?\<chi> J.TT"
         using D.mkCone_def by simp
       show "span (pr1 a b) (pr0 a b)"
         using 1 \<chi>.preserves_reflects_arr J.seqE J.arr_char J.seq_char J.is_category
               D.is_rendered_commutative_by_cone \<chi>.cone_axioms
         by metis
       show "cod (pr1 a b) = a"
         using 1 \<chi>.preserves_cod [of J.FF] J.cod_char J.arr_char by auto
       show "cod (pr0 a b) = b"
         using 1 \<chi>.preserves_cod [of J.TT] J.cod_char J.arr_char by auto
       next
       fix f g
       assume fg: "span f g"
       show "\<exists>!l. pr1 (cod f) (cod g) \<cdot> l = f \<and> pr0 (cod f) (cod g) \<cdot> l = g"
       proof -
         interpret J: binary_product_shape .
         interpret D: binary_product_diagram C \<open>cod f\<close> \<open>cod g\<close>
           using fg by unfold_locales auto
         let ?\<chi> = "D.mkCone (pr1 (cod f) (cod g)) (pr0 (cod f) (cod g))"
         interpret \<chi>: limit_cone J.comp C D.map \<open>dom (pr1 (cod f) (cod g))\<close> ?\<chi>
           using fg pr_yields_binary_product [of "cod f" "cod g"] has_as_binary_product_def
           by simp
         interpret \<chi>: binary_product_cone C \<open>cod f\<close> \<open>cod g\<close>
                        \<open>pr1 (cod f) (cod g)\<close> \<open>pr0 (cod f) (cod g)\<close> ..
         have 1: "pr1 (cod f) (cod g) = ?\<chi> J.FF \<and> pr0 (cod f) (cod g) = ?\<chi> J.TT"
           using D.mkCone_def by simp
         show "\<exists>!l. pr1 (cod f) (cod g) \<cdot> l = f \<and> pr0 (cod f) (cod g) \<cdot> l = g"
         proof -
           have "\<exists>!l. \<guillemotleft>l : dom f \<rightarrow> dom (pr1 (cod f) (cod g))\<guillemotright> \<and>
                      pr1 (cod f) (cod g) \<cdot> l = f \<and> pr0 (cod f) (cod g) \<cdot> l = g"
             using fg \<chi>.is_universal' by simp
           moreover have "\<And>l. pr1 (cod f) (cod g) \<cdot> l = f
                                 \<Longrightarrow> \<guillemotleft>l : dom f \<rightarrow> dom (pr1 (cod f) (cod g))\<guillemotright>"
             using fg dom_comp in_homI seqE seqI by metis
           ultimately show ?thesis by auto
         qed
       qed
     qed
 
     proposition extends_to_elementary_category_with_binary_products:
     shows "elementary_category_with_binary_products C pr0 pr1"
       ..
 
   end
 
   context elementary_category_with_binary_products
   begin
 
     interpretation category_with_binary_products C
     proof
       show "has_binary_products"
       proof (unfold has_binary_products_def)
         have "\<And>a b. ide a \<and> ide b \<Longrightarrow> \<exists>p0 p1. has_as_binary_product a b p0 p1"
         proof -
           fix a b
           assume ab: "ide a \<and> ide b"
           interpret J: binary_product_shape .
           interpret D: binary_product_diagram C a b
             using ab by unfold_locales auto
           have 2: "D.is_rendered_commutative_by \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]"
             using ab by simp
           let ?\<chi> = "D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]"
           interpret \<chi>: cone J.comp C D.map \<open>dom \<pp>\<^sub>1[a, b]\<close> ?\<chi>
             using D.cone_mkCone 2 by auto
           interpret \<chi>: limit_cone J.comp C D.map \<open>dom \<pp>\<^sub>1[a, b]\<close> ?\<chi>
           proof
             fix a' \<chi>'
             assume \<chi>': "D.cone a' \<chi>'"
             interpret \<chi>': cone J.comp C D.map a' \<chi>'
               using \<chi>' by simp
             show "\<exists>!h. \<guillemotleft>h : a' \<rightarrow> dom \<pp>\<^sub>1[a, b]\<guillemotright> \<and>
                        D.cones_map h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]) = \<chi>'"
             proof
               let ?h = "\<langle>\<chi>' J.FF, \<chi>' J.TT\<rangle>"
               show h': "\<guillemotleft>?h : a' \<rightarrow> dom \<pp>\<^sub>1[a, b]\<guillemotright> \<and>
                         D.cones_map ?h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]) = \<chi>'"
               proof
                 show h: "\<guillemotleft>?h : a' \<rightarrow> dom \<pp>\<^sub>1[a, b]\<guillemotright>"
                   using ab tuple_in_hom J.ide_char by fastforce
                 show "D.cones_map ?h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]) = \<chi>'"
                 proof -
                   interpret \<chi>'h: cone J.comp C D.map a'
                                    \<open>D.cones_map ?h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b])\<close>
                   proof -
                     have "D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b] \<in> D.cones (cod \<langle>\<chi>' J.FF, \<chi>' J.TT\<rangle>)"
                       using ab h D.cone_mkCone D.is_rendered_commutative_by_cone
                             \<chi>.cone_axioms
                       by auto
                     hence "D.cones_map ?h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]) \<in> D.cones a'"
                       using ab h D.cones_map_mapsto by blast
                     thus "D.cone a' (D.cones_map ?h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]))"
                       by simp
                   qed
                   show ?thesis
                   proof -
                     have "\<And>j. J.ide j \<Longrightarrow> D.cones_map ?h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]) j = \<chi>' j"
                       using ab h J.ide_char D.mkCone_def \<chi>.cone_axioms by auto
                     thus ?thesis
                       using NaturalTransformation.eqI
                             \<chi>'.natural_transformation_axioms \<chi>'h.natural_transformation_axioms
                       by blast
                   qed
                 qed
               qed
               show "\<And>h. \<guillemotleft>h : a' \<rightarrow> dom \<pp>\<^sub>1[a, b]\<guillemotright> \<and>
                         D.cones_map h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]) = \<chi>' \<Longrightarrow> h = ?h"
               proof -
                 fix h
                 assume 1: "\<guillemotleft>h : a' \<rightarrow> dom \<pp>\<^sub>1[a, b]\<guillemotright> \<and>
                            D.cones_map h (D.mkCone \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]) = \<chi>'"
                 hence "cod h = dom \<pp>\<^sub>1[a, b]" by auto
                 show "h = ?h"
                   using 1 ab \<chi>.cone_axioms D.mkCone_def h' pr_joint_monic [of a b h ?h]
                   by auto
               qed
             qed
           qed
           have "has_as_binary_product a b \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]"
             using ab has_as_binary_product_def \<chi>.limit_cone_axioms by blast
           thus "\<exists>p0 p1. has_as_binary_product a b p0 p1"
             by blast
         qed
         thus "\<forall>a b. ide a \<and> ide b \<longrightarrow> (\<exists>p0 p1. has_as_binary_product a b p0 p1)"
           by simp
       qed
     qed
 
     proposition is_category_with_binary_products:
     shows "category_with_binary_products C"
       ..
 
   end
 
   subsection "Further Properties"
 
   context elementary_category_with_binary_products
   begin
 
     lemma interchange:
     assumes "seq h f" and "seq k g"
     shows "(h \<otimes> k) \<cdot> (f \<otimes> g) = h \<cdot> f \<otimes> k \<cdot> g"
       using assms prod_def comp_tuple_arr comp_assoc by fastforce
 
     lemma pr_naturality [simp]:
     assumes "arr g" and "dom g = b" and "cod g = d"
         and "arr f" and "dom f = a" and "cod f = c"
     shows "\<pp>\<^sub>0[c, d] \<cdot> (f \<otimes> g) = g \<cdot> \<pp>\<^sub>0[a, b]"
     and "\<pp>\<^sub>1[c, d] \<cdot> (f \<otimes> g) = f \<cdot> \<pp>\<^sub>1[a, b]"
       using assms prod_def by fastforce+
 
     abbreviation dup ("\<d>[_]")
     where "\<d>[f] \<equiv> \<langle>f, f\<rangle>"
 
     lemma dup_in_hom [intro, simp]:
     assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
     shows "\<guillemotleft>\<d>[f] : a \<rightarrow> b \<otimes> b\<guillemotright>"
       using assms by fastforce
 
     lemma dup_simps [simp]:
     assumes "arr f"
     shows "arr \<d>[f]" and "dom \<d>[f] = dom f" and "cod \<d>[f] = cod f \<otimes> cod f"
       using assms dup_in_hom by auto
 
     lemma dup_naturality:
     assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
     shows "\<d>[b] \<cdot> f = (f \<otimes> f) \<cdot> \<d>[a]"
       using assms prod_def comp_arr_dom comp_cod_arr comp_tuple_arr comp_assoc
       by fastforce
 
     lemma pr_dup [simp]:
     assumes "ide a"
     shows "\<pp>\<^sub>0[a, a] \<cdot> \<d>[a] = a" and "\<pp>\<^sub>1[a, a] \<cdot> \<d>[a] = a"
       using assms by simp_all
 
     lemma prod_tuple:
     assumes "span f g" and "seq h f" and "seq k g"
     shows "(h \<otimes> k) \<cdot> \<langle>f, g\<rangle> = \<langle>h \<cdot> f, k \<cdot> g\<rangle>"
       using assms prod_def comp_assoc comp_tuple_arr by fastforce
 
     lemma tuple_eqI:
     assumes "seq \<pp>\<^sub>0[b, c] f" and "seq \<pp>\<^sub>1[b, c] f"
     and "\<pp>\<^sub>0[b, c] \<cdot> f = f0" and "\<pp>\<^sub>1[b, c] \<cdot> f = f1"
     shows "f = \<langle>f1, f0\<rangle>"
       using assms pr_joint_monic [of b c "\<langle>f1, f0\<rangle>" f] pr_tuple by auto
 
     definition assoc ("\<a>[_, _, _]")
     where "\<a>[a, b, c] \<equiv> \<langle>\<pp>\<^sub>1[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c], \<langle>\<pp>\<^sub>0[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c], \<pp>\<^sub>0[a \<otimes> b, c]\<rangle>\<rangle>"
 
     definition assoc' ("\<a>\<^sup>-\<^sup>1[_, _, _]")
     where "\<a>\<^sup>-\<^sup>1[a, b, c] \<equiv> \<langle>\<langle>\<pp>\<^sub>1[a, b \<otimes> c], \<pp>\<^sub>1[b, c] \<cdot> \<pp>\<^sub>0[a, b \<otimes> c]\<rangle>, \<pp>\<^sub>0[b, c] \<cdot> \<pp>\<^sub>0[a, b \<otimes> c]\<rangle>"
 
     lemma assoc_in_hom [intro]:
     assumes "ide a" and "ide b" and "ide c"
     shows "\<guillemotleft>\<a>[a, b, c] : (a \<otimes> b) \<otimes> c \<rightarrow> a \<otimes> (b \<otimes> c)\<guillemotright>"
       using assms assoc_def by auto
 
     lemma assoc_simps [simp]:
     assumes "ide a" and "ide b" and "ide c"
     shows "arr \<a>[a, b, c]"
     and "dom \<a>[a, b, c] = (a \<otimes> b) \<otimes> c"
     and "cod \<a>[a, b, c] = a \<otimes> (b \<otimes> c)"
       using assms assoc_in_hom by auto
 
     lemma assoc'_in_hom [intro]:
     assumes "ide a" and "ide b" and "ide c"
     shows "\<guillemotleft>\<a>\<^sup>-\<^sup>1[a, b, c] : a \<otimes> (b \<otimes> c) \<rightarrow> (a \<otimes> b) \<otimes> c\<guillemotright>"
       using assms assoc'_def by auto
 
     lemma assoc'_simps [simp]:
     assumes "ide a" and "ide b" and "ide c"
     shows "arr \<a>\<^sup>-\<^sup>1[a, b, c]"
     and "dom \<a>\<^sup>-\<^sup>1[a, b, c] = a \<otimes> (b \<otimes> c)"
     and "cod \<a>\<^sup>-\<^sup>1[a, b, c] = (a \<otimes> b) \<otimes> c"
       using assms assoc'_in_hom by auto
 
     lemma assoc_naturality:
     assumes "\<guillemotleft>f0 : a0 \<rightarrow> b0\<guillemotright>" and "\<guillemotleft>f1 : a1 \<rightarrow> b1\<guillemotright>" and "\<guillemotleft>f2 : a2 \<rightarrow> b2\<guillemotright>"
     shows "\<a>[b0, b1, b2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2) = (f0 \<otimes> (f1 \<otimes> f2)) \<cdot> \<a>[a0, a1, a2]"
     proof -
       have "\<pp>\<^sub>0[b0, b1 \<otimes> b2] \<cdot> \<a>[b0, b1, b2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2) =
             \<pp>\<^sub>0[b0, b1 \<otimes> b2] \<cdot> (f0 \<otimes> (f1 \<otimes> f2)) \<cdot> \<a>[a0, a1, a2]"
       proof -
         have "\<pp>\<^sub>0[b0, b1 \<otimes> b2] \<cdot> \<a>[b0, b1, b2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2) =
               (\<pp>\<^sub>0[b0, b1 \<otimes> b2] \<cdot> \<a>[b0, b1, b2]) \<cdot> ((f0 \<otimes> f1) \<otimes> f2)"
           using comp_assoc by simp
         also have "... = \<langle>\<pp>\<^sub>0[b0, b1] \<cdot> \<pp>\<^sub>1[b0 \<otimes> b1, b2], \<pp>\<^sub>0[b0 \<otimes> b1, b2]\<rangle> \<cdot> ((f0 \<otimes> f1) \<otimes> f2)"
           using assms assoc_def by fastforce
         also have "... = \<langle>(\<pp>\<^sub>0[b0, b1] \<cdot> \<pp>\<^sub>1[b0 \<otimes> b1, b2]) \<cdot> ((f0 \<otimes> f1) \<otimes> f2),
                           \<pp>\<^sub>0[b0 \<otimes> b1, b2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2)\<rangle>"
           using assms comp_tuple_arr by fastforce
         also have "... = \<langle>(\<pp>\<^sub>0[b0, b1] \<cdot> (f0 \<otimes> f1)) \<cdot> \<pp>\<^sub>1[a0 \<otimes> a1, a2], f2 \<cdot> \<pp>\<^sub>0[a0 \<otimes> a1, a2]\<rangle>"
           using assms comp_assoc by fastforce
         also have "... = \<langle>f1 \<cdot> \<pp>\<^sub>0[a0, a1] \<cdot> \<pp>\<^sub>1[a0 \<otimes> a1, a2], f2 \<cdot> \<pp>\<^sub>0[a0 \<otimes> a1, a2]\<rangle>"
           using assms comp_assoc
           by (metis in_homE pr_naturality(1))
         also have "... = \<pp>\<^sub>0[b0, b1 \<otimes> b2] \<cdot> (f0 \<otimes> (f1 \<otimes> f2)) \<cdot> \<a>[a0, a1, a2]"
           using assms comp_assoc assoc_def prod_tuple by fastforce
         finally show ?thesis by blast
       qed
       moreover have "\<pp>\<^sub>1[b0, b1 \<otimes> b2] \<cdot> \<a>[b0, b1, b2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2) =
                      \<pp>\<^sub>1[b0, b1 \<otimes> b2] \<cdot> (f0 \<otimes> (f1 \<otimes> f2)) \<cdot> \<a>[a0, a1, a2]"
       proof -
         have "\<pp>\<^sub>1[b0, b1 \<otimes> b2] \<cdot> \<a>[b0, b1, b2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2) =
               (\<pp>\<^sub>1[b0, b1 \<otimes> b2] \<cdot> \<a>[b0, b1, b2]) \<cdot> ((f0 \<otimes> f1) \<otimes> f2)"
           using comp_assoc by simp
         also have "... = (\<pp>\<^sub>1[b0, b1] \<cdot> \<pp>\<^sub>1[b0 \<otimes> b1, b2]) \<cdot> ((f0 \<otimes> f1) \<otimes> f2)"
           using assms assoc_def by fastforce
         also have "... = (\<pp>\<^sub>1[b0, b1] \<cdot> (f0 \<otimes> f1)) \<cdot> \<pp>\<^sub>1[a0 \<otimes> a1, a2]"
           using assms comp_assoc by fastforce
         also have "... = f0 \<cdot> \<pp>\<^sub>1[a0, a1] \<cdot> \<pp>\<^sub>1[a0 \<otimes> a1, a2]"
           using assms comp_assoc
           by (metis in_homE pr_naturality(2))
         also have "... = \<pp>\<^sub>1[b0, b1 \<otimes> b2] \<cdot> (f0 \<otimes> (f1 \<otimes> f2)) \<cdot> \<a>[a0, a1, a2]"
         proof -
           have "\<pp>\<^sub>1[b0, b1 \<otimes> b2] \<cdot> (f0 \<otimes> (f1 \<otimes> f2)) \<cdot> \<a>[a0, a1, a2] =
                 (\<pp>\<^sub>1[b0, b1 \<otimes> b2] \<cdot> (f0 \<otimes> (f1 \<otimes> f2))) \<cdot> \<a>[a0, a1, a2]"
             using comp_assoc by simp
           also have "... = f0 \<cdot> \<pp>\<^sub>1[a0, a1 \<otimes> a2] \<cdot> \<a>[a0, a1, a2]"
             using assms comp_assoc by fastforce
           also have "... = f0 \<cdot> \<pp>\<^sub>1[a0, a1] \<cdot> \<pp>\<^sub>1[a0 \<otimes> a1, a2]"
             using assms assoc_def by fastforce
           finally show ?thesis by simp
         qed
         finally show ?thesis by blast
       qed
       ultimately show ?thesis
         using assms pr_joint_monic [of b0 "b1 \<otimes> b2" "\<a>[b0, b1, b2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2)"
                                        "(f0 \<otimes> (f1 \<otimes> f2)) \<cdot> \<a>[a0, a1, a2]"]
         by fastforce
     qed
 
     lemma pentagon:
     assumes "ide a" and "ide b" and "ide c" and "ide d"
     shows "((a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d) = \<a>[a, b, c \<otimes> d] \<cdot> \<a>[a \<otimes> b, c, d]"
     proof (intro pr_joint_monic
                    [where h = "((a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d)"
                       and h' = "\<a>[a, b, c \<otimes> d] \<cdot> \<a>[a \<otimes> b, c, d]"])
       show "seq \<pp>\<^sub>0[a, b \<otimes> (c \<otimes> d)] (((a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d))"
         using assms by simp
       show "\<pp>\<^sub>1[a, b \<otimes> c \<otimes> d] \<cdot> ((a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d) =
             \<pp>\<^sub>1[a, b \<otimes> c \<otimes> d] \<cdot> \<a>[a, b, c \<otimes> d] \<cdot> \<a>[a \<otimes> b, c, d]"
       proof -
         have "\<pp>\<^sub>1[a, b \<otimes> c \<otimes> d] \<cdot> ((a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d) =
               ((\<pp>\<^sub>1[a, b \<otimes> c \<otimes> d] \<cdot> (a \<otimes> \<a>[b, c, d])) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d)"
           using comp_assoc by simp
         also have "... = (\<pp>\<^sub>1[a, (b \<otimes> c) \<otimes> d] \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d)"
           using assms pr_naturality(2) comp_cod_arr by force
         also have "... = \<pp>\<^sub>1[a, b \<otimes> c] \<cdot> \<pp>\<^sub>1[a \<otimes> b \<otimes> c, d] \<cdot> (\<a>[a, b, c] \<otimes> d)"
           using assms assoc_def comp_assoc by simp
         also have "... = (\<pp>\<^sub>1[a, b \<otimes> c] \<cdot> \<a>[a, b, c]) \<cdot> \<pp>\<^sub>1[(a \<otimes> b) \<otimes> c, d]"
           using assms pr_naturality(2) comp_assoc by fastforce
         also have "... = \<pp>\<^sub>1[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c] \<cdot> \<pp>\<^sub>1[(a \<otimes> b) \<otimes> c, d]"
           using assms assoc_def comp_assoc by simp
         finally have "\<pp>\<^sub>1[a, b \<otimes> c \<otimes> d] \<cdot> ((a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d) =
                       \<pp>\<^sub>1[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c] \<cdot> \<pp>\<^sub>1[(a \<otimes> b) \<otimes> c, d]"
           by blast
         also have "... = \<pp>\<^sub>1[a, b \<otimes> c \<otimes> d] \<cdot> \<a>[a, b, c \<otimes> d] \<cdot> \<a>[a \<otimes> b, c, d]"
           using assms assoc_def comp_assoc by auto
         finally show ?thesis by blast
       qed
       show "\<pp>\<^sub>0[a, b \<otimes> (c \<otimes> d)] \<cdot> ((a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d) =
             \<pp>\<^sub>0[a, b \<otimes> (c \<otimes> d)] \<cdot> \<a>[a, b, c \<otimes> d] \<cdot> \<a>[a \<otimes> b, c, d]"
       proof -
         have "\<pp>\<^sub>0[a, b \<otimes> (c \<otimes> d)] \<cdot> ((a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d]) \<cdot> (\<a>[a, b, c] \<otimes> d) =
               \<pp>\<^sub>0[a, b \<otimes> c \<otimes> d] \<cdot>
                 ((a \<otimes> \<langle>\<pp>\<^sub>1[b, c] \<cdot> \<pp>\<^sub>1[b \<otimes> c, d], \<langle>\<pp>\<^sub>0[b, c] \<cdot> \<pp>\<^sub>1[b \<otimes> c, d], \<pp>\<^sub>0[b \<otimes> c, d]\<rangle>\<rangle>) \<cdot>
                  \<langle>\<pp>\<^sub>1[a, b \<otimes> c] \<cdot> \<pp>\<^sub>1[a \<otimes> b \<otimes> c, d],
                   \<langle>\<pp>\<^sub>0[a, b \<otimes> c] \<cdot> \<pp>\<^sub>1[a \<otimes> b \<otimes> c, d], \<pp>\<^sub>0[a \<otimes> b \<otimes> c, d]\<rangle>\<rangle>) \<cdot>
                 (\<langle>\<pp>\<^sub>1[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c], \<langle>\<pp>\<^sub>0[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c], \<pp>\<^sub>0[a \<otimes> b, c]\<rangle>\<rangle> \<otimes> d)"
           using assms assoc_def by simp
         also have "... = \<langle>\<pp>\<^sub>1[b, c] \<cdot> \<pp>\<^sub>1[b \<otimes> c, d],
                           \<langle>\<pp>\<^sub>0[b, c] \<cdot> \<pp>\<^sub>1[b \<otimes> c, d], \<pp>\<^sub>0[b \<otimes> c, d]\<rangle>\<rangle> \<cdot> (\<pp>\<^sub>0[a, (b \<otimes> c) \<otimes> d] \<cdot>
                             \<langle>\<pp>\<^sub>1[a, b \<otimes> c] \<cdot> \<pp>\<^sub>1[a \<otimes> b \<otimes> c, d],
                              \<langle>\<pp>\<^sub>0[a, b \<otimes> c] \<cdot> \<pp>\<^sub>1[a \<otimes> b \<otimes> c, d], \<pp>\<^sub>0[a \<otimes> b \<otimes> c, d]\<rangle>\<rangle>) \<cdot>
                             (\<langle>\<pp>\<^sub>1[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c],
                               \<langle>\<pp>\<^sub>0[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c], \<pp>\<^sub>0[a \<otimes> b, c]\<rangle>\<rangle> \<otimes> d)"
         proof -
           have "\<pp>\<^sub>0[a, b \<otimes> c \<otimes> d] \<cdot>
                   (a \<otimes> \<langle>\<pp>\<^sub>1[b, c] \<cdot> \<pp>\<^sub>1[b \<otimes> c, d], \<langle>\<pp>\<^sub>0[b, c] \<cdot> \<pp>\<^sub>1[b \<otimes> c, d], \<pp>\<^sub>0[b \<otimes> c, d]\<rangle>\<rangle>) =
                 \<langle>\<pp>\<^sub>1[b, c] \<cdot> \<pp>\<^sub>1[b \<otimes> c, d], \<langle>\<pp>\<^sub>0[b, c] \<cdot> \<pp>\<^sub>1[b \<otimes> c, d], \<pp>\<^sub>0[b \<otimes> c, d]\<rangle>\<rangle> \<cdot>
                   \<pp>\<^sub>0[a, (b \<otimes> c) \<otimes> d]"
             using assms assoc_def ide_in_hom pr_naturality(1) by auto
           thus ?thesis using comp_assoc by metis
         qed
         also have "... = \<langle>\<pp>\<^sub>0[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c] \<cdot> \<pp>\<^sub>1[(a \<otimes> b) \<otimes> c, d],
                           \<langle>\<pp>\<^sub>0[a \<otimes> b, c] \<cdot> \<pp>\<^sub>1[(a \<otimes> b) \<otimes> c, d], d \<cdot> \<pp>\<^sub>0[(a \<otimes> b) \<otimes> c, d]\<rangle>\<rangle>"
           using assms comp_assoc by simp
         also have "... = \<langle>\<pp>\<^sub>0[a, b] \<cdot> \<pp>\<^sub>1[a \<otimes> b, c] \<cdot> \<pp>\<^sub>1[(a \<otimes> b) \<otimes> c, d],
                           \<langle>\<pp>\<^sub>0[a \<otimes> b, c] \<cdot> \<pp>\<^sub>1[(a \<otimes> b) \<otimes> c, d], \<pp>\<^sub>0[(a \<otimes> b) \<otimes> c, d]\<rangle>\<rangle>"
           using assms comp_cod_arr by simp
         also have "... = \<pp>\<^sub>0[a, b \<otimes> (c \<otimes> d)] \<cdot> \<a>[a, b, c \<otimes> d] \<cdot> \<a>[a \<otimes> b, c, d]"
           using assms assoc_def comp_assoc by simp
         finally show ?thesis by simp
       qed
     qed
 
     lemma inverse_arrows_assoc:
     assumes "ide a" and "ide b" and "ide c"
     shows "inverse_arrows \<a>[a, b, c] \<a>\<^sup>-\<^sup>1[a, b, c]"
       using assms assoc_def assoc'_def comp_assoc
       by (auto simp add: tuple_pr_arr)
 
     interpretation CC: product_category C C ..
 
     abbreviation Prod
     where "Prod fg \<equiv> fst fg \<otimes> snd fg"
     abbreviation Prod'
     where "Prod' fg \<equiv> snd fg \<otimes> fst fg"
 
     interpretation \<Pi>: binary_functor C C C Prod
       using tuple_ext CC.comp_char interchange
       apply unfold_locales
           apply auto
       by (metis prod_def seqE)+
 
     interpretation Prod': binary_functor C C C Prod'
       using tuple_ext CC.comp_char interchange
       apply unfold_locales
           apply auto
       by (metis prod_def seqE)+
 
     lemma binary_functor_Prod:
     shows "binary_functor C C C Prod" and "binary_functor C C C Prod'"
       ..
 
     definition sym ("\<s>[_, _]")
     where "\<s>[a1, a0] \<equiv> if ide a0 \<and> ide a1 then \<langle>\<pp>\<^sub>0[a1, a0], \<pp>\<^sub>1[a1, a0]\<rangle> else null"
 
     lemma sym_in_hom [intro]:
     assumes "ide a" and "ide b"
     shows "\<guillemotleft>\<s>[a, b] : a \<otimes> b \<rightarrow> b \<otimes> a\<guillemotright>"
       using assms sym_def by auto
 
     lemma sym_simps [simp]:
     assumes "ide a" and "ide b"
     shows "arr \<s>[a, b]" and "dom \<s>[a, b] = a \<otimes> b" and "cod \<s>[a, b] = b \<otimes> a"
       using assms sym_in_hom by auto
 
     lemma comp_sym_tuple [simp]:
     assumes "\<guillemotleft>f0 : a \<rightarrow> b0\<guillemotright>" and "\<guillemotleft>f1 : a \<rightarrow> b1\<guillemotright>"
     shows "\<s>[b0, b1] \<cdot> \<langle>f0, f1\<rangle> = \<langle>f1, f0\<rangle>"
       using assms sym_def comp_tuple_arr by fastforce
 
     lemma prj_sym [simp]:
     assumes "ide a0" and "ide a1"
     shows "\<pp>\<^sub>0[a1, a0] \<cdot> \<s>[a0, a1] = \<pp>\<^sub>1[a0, a1]"
     and "\<pp>\<^sub>1[a1, a0] \<cdot> \<s>[a0, a1] = \<pp>\<^sub>0[a0, a1]"
       using assms sym_def by auto
 
     lemma comp_sym_sym [simp]:
     assumes "ide a0" and "ide a1"
     shows "\<s>[a1, a0] \<cdot> \<s>[a0, a1] = (a0 \<otimes> a1)"
       using assms sym_def comp_tuple_arr by auto
 
     lemma sym_inverse_arrows:
     assumes "ide a0" and "ide a1"
     shows "inverse_arrows \<s>[a0, a1] \<s>[a1, a0]"
       using assms sym_in_hom comp_sym_sym by auto
 
     lemma sym_assoc_coherence:
     assumes "ide a" and "ide b" and "ide c"
     shows "\<a>[b, c, a] \<cdot> \<s>[a, b \<otimes> c] \<cdot> \<a>[a, b, c] = (b \<otimes> \<s>[a, c]) \<cdot> \<a>[b, a, c] \<cdot> (\<s>[a, b] \<otimes> c)"
       using assms sym_def assoc_def comp_assoc prod_tuple comp_cod_arr by simp
 
     lemma sym_naturality:
     assumes "\<guillemotleft>f0 : a0 \<rightarrow> b0\<guillemotright>" and "\<guillemotleft>f1 : a1 \<rightarrow> b1\<guillemotright>"
     shows "\<s>[b0, b1] \<cdot> (f0 \<otimes> f1) = (f1 \<otimes> f0) \<cdot> \<s>[a0, a1]"
       using assms sym_def comp_assoc prod_tuple by fastforce
 
     abbreviation \<sigma>
     where "\<sigma> fg \<equiv> \<s>[cod (fst fg), cod (snd fg)] \<cdot> (fst fg \<otimes> snd fg)"
 
     interpretation \<sigma>: natural_transformation CC.comp C Prod Prod' \<sigma>
       using sym_def CC.arr_char CC.null_char comp_arr_dom comp_cod_arr
       apply unfold_locales
           apply auto
       using arr_cod_iff_arr ideD(1)
         apply metis
       using arr_cod_iff_arr ideD(1)
        apply metis
       using prod_tuple by simp
 
     lemma \<sigma>_is_natural_transformation:
     shows "natural_transformation CC.comp C Prod Prod' \<sigma>"
       ..
 
     abbreviation Diag
     where "Diag f \<equiv> if arr f then (f, f) else CC.null"
 
     interpretation \<Delta>: "functor" C CC.comp Diag
       by (unfold_locales, auto)
 
     lemma functor_Diag:
     shows "functor C CC.comp Diag"
       ..
 
     interpretation \<Delta>o\<Pi>: composite_functor CC.comp C CC.comp Prod Diag ..
     interpretation \<Pi>o\<Delta>: composite_functor C CC.comp C Diag Prod ..
 
     abbreviation \<pi>
     where "\<pi> \<equiv> \<lambda>(f, g). (\<pp>\<^sub>1[cod f, cod g] \<cdot> (f \<otimes> g), \<pp>\<^sub>0[cod f, cod g] \<cdot> (f \<otimes> g))"
 
     interpretation \<pi>: transformation_by_components CC.comp CC.comp \<Delta>o\<Pi>.map CC.map \<pi>
       using pr_naturality comp_arr_dom comp_cod_arr
       by unfold_locales auto
 
     lemma \<pi>_is_natural_transformation:
     shows "natural_transformation CC.comp CC.comp \<Delta>o\<Pi>.map CC.map \<pi>"
     proof -
       have "\<pi>.map = \<pi>"
         using \<pi>.map_def ext \<Pi>.is_extensional comp_arr_dom comp_cod_arr by auto
       thus "natural_transformation CC.comp CC.comp \<Delta>o\<Pi>.map CC.map \<pi>"
         using \<pi>.natural_transformation_axioms by simp
     qed
 
     interpretation \<delta>: natural_transformation C C map \<Pi>o\<Delta>.map dup
       using dup_naturality comp_arr_dom comp_cod_arr prod_tuple tuple_ext
       by unfold_locales auto
 
     lemma dup_is_natural_transformation:
     shows "natural_transformation C C map \<Pi>o\<Delta>.map dup"
       ..
 
     interpretation \<Delta>o\<Pi>o\<Delta>: composite_functor C CC.comp CC.comp Diag \<Delta>o\<Pi>.map ..
     interpretation \<Pi>o\<Delta>o\<Pi>: composite_functor CC.comp C C Prod \<Pi>o\<Delta>.map ..
 
     interpretation \<Delta>o\<delta>: natural_transformation C CC.comp Diag \<Delta>o\<Pi>o\<Delta>.map \<open>Diag \<circ> dup\<close>
     proof -
       have "Diag \<circ> map = Diag"
         by auto
       thus "natural_transformation C CC.comp Diag \<Delta>o\<Pi>o\<Delta>.map (Diag \<circ> dup)"
         using \<Delta>.natural_transformation_axioms \<delta>.natural_transformation_axioms o_assoc
               horizontal_composite [of C C map \<Pi>o\<Delta>.map dup CC.comp Diag Diag Diag]
         by metis
     qed
 
     interpretation \<delta>o\<Pi>: natural_transformation CC.comp C Prod \<Pi>o\<Delta>o\<Pi>.map \<open>dup \<circ> Prod\<close>
       using \<delta>.natural_transformation_axioms \<Pi>.natural_transformation_axioms o_assoc
             horizontal_composite [of CC.comp C Prod Prod Prod C map \<Pi>o\<Delta>.map dup]
       by simp
 
     interpretation \<pi>o\<Delta>: natural_transformation C CC.comp \<Delta>o\<Pi>o\<Delta>.map Diag \<open>\<pi>.map \<circ> Diag\<close>
       using \<pi>.natural_transformation_axioms \<Delta>.natural_transformation_axioms
             horizontal_composite
               [of C CC.comp Diag Diag Diag CC.comp \<Delta>o\<Pi>.map CC.map \<pi>.map]
       by simp
 
     interpretation \<Pi>o\<pi>: natural_transformation CC.comp C \<Pi>o\<Delta>o\<Pi>.map Prod \<open>Prod \<circ> \<pi>.map\<close>
     proof -
       have "Prod \<circ> \<Delta>o\<Pi>.map = \<Pi>o\<Delta>o\<Pi>.map"
         by auto
       thus "natural_transformation CC.comp C \<Pi>o\<Delta>o\<Pi>.map Prod (Prod \<circ> \<pi>.map)"
         using \<pi>.natural_transformation_axioms \<Pi>.natural_transformation_axioms o_assoc
               horizontal_composite
                 [of CC.comp CC.comp \<Delta>o\<Pi>.map CC.map \<pi>.map C Prod Prod Prod]
         by simp
     qed
 
     interpretation \<Delta>o\<delta>_\<pi>o\<Delta>: vertical_composite C CC.comp Diag \<Delta>o\<Pi>o\<Delta>.map Diag
                                \<open>Diag \<circ> dup\<close> \<open>\<pi>.map \<circ> Diag\<close>
       ..
     interpretation \<Pi>o\<pi>_\<delta>o\<Pi>: vertical_composite CC.comp C Prod \<Pi>o\<Delta>o\<Pi>.map Prod
                                \<open>dup \<circ> Prod\<close> \<open>Prod \<circ> \<pi>.map\<close>
       ..
 
     interpretation \<Delta>\<Pi>: unit_counit_adjunction CC.comp C Diag Prod dup \<pi>.map
     proof
       show "\<Delta>o\<delta>_\<pi>o\<Delta>.map = Diag"
       proof
         fix f
         have "\<not> arr f \<Longrightarrow> \<Delta>o\<delta>_\<pi>o\<Delta>.map f = Diag f"
           by (simp add: \<Delta>o\<delta>_\<pi>o\<Delta>.is_extensional)
         moreover have "arr f \<Longrightarrow> \<Delta>o\<delta>_\<pi>o\<Delta>.map f = Diag f"
           using comp_cod_arr comp_assoc \<Delta>o\<delta>_\<pi>o\<Delta>.map_def by auto
         ultimately show "\<Delta>o\<delta>_\<pi>o\<Delta>.map f = Diag f" by blast
       qed
       show "\<Pi>o\<pi>_\<delta>o\<Pi>.map = Prod"
       proof
         fix fg
         show "\<Pi>o\<pi>_\<delta>o\<Pi>.map fg = Prod fg"
         proof -
           have "\<not> CC.arr fg \<Longrightarrow> ?thesis"
             by (simp add: \<Pi>.is_extensional \<Pi>o\<pi>_\<delta>o\<Pi>.is_extensional)
           moreover have "CC.arr fg \<Longrightarrow> ?thesis"
           proof -
             assume fg: "CC.arr fg"
             have 1: "dup (Prod fg) = \<langle>cod (fst fg) \<otimes> cod (snd fg), cod (fst fg) \<otimes> cod (snd fg)\<rangle> \<cdot>
                                         (fst fg \<otimes> snd fg)"
               using fg \<delta>.is_natural_2
               apply simp
               by (metis (no_types, lifting) prod_simps(1) prod_simps(3))
             have "\<Pi>o\<pi>_\<delta>o\<Pi>.map fg =
                   (\<pp>\<^sub>1[cod (fst fg), cod (snd fg)] \<otimes> \<pp>\<^sub>0[cod (fst fg), cod (snd fg)]) \<cdot>
                     \<langle>cod (fst fg) \<otimes> cod (snd fg), cod (fst fg) \<otimes> cod (snd fg)\<rangle> \<cdot>
                     (fst fg \<otimes> snd fg)"
               using fg 1 \<Pi>o\<pi>_\<delta>o\<Pi>.map_def comp_cod_arr by simp
             also have "... = ((\<pp>\<^sub>1[cod (fst fg), cod (snd fg)] \<otimes> \<pp>\<^sub>0[cod (fst fg), cod (snd fg)]) \<cdot>
                               \<langle>cod (fst fg) \<otimes> cod (snd fg), cod (fst fg) \<otimes> cod (snd fg)\<rangle>) \<cdot>
                              (fst fg \<otimes> snd fg)"
               using comp_assoc by simp
             also have "... = \<langle>\<pp>\<^sub>1[cod (fst fg), cod (snd fg)] \<cdot> (cod (fst fg) \<otimes> cod (snd fg)),
                               \<pp>\<^sub>0[cod (fst fg), cod (snd fg)] \<cdot> (cod (fst fg) \<otimes> cod (snd fg))\<rangle> \<cdot>
                              (fst fg \<otimes> snd fg)"
               using fg prod_tuple by simp
             also have "... = Prod fg"
               using fg comp_arr_dom \<Pi>.is_natural_2 by auto
             finally show ?thesis by simp
           qed
           ultimately show ?thesis by blast
         qed
       qed
     qed
 
     proposition induces_unit_counit_adjunction:
     shows "unit_counit_adjunction CC.comp C Diag Prod dup \<pi>.map"
       using \<Delta>\<Pi>.unit_counit_adjunction_axioms by simp
 
   end
 
   section "Category with Terminal Object"
 
   locale category_with_terminal_object =
     category +
   assumes has_terminal: "\<exists>t. terminal t"
 
   locale elementary_category_with_terminal_object =
     category C
   for C :: "'a comp"                              (infixr "\<cdot>" 55)
   and one :: "'a"                                 ("\<one>")
   and trm :: "'a \<Rightarrow> 'a"                           ("\<t>[_]") +
   assumes ide_one: "ide \<one>"
   and trm_in_hom: "ide a \<Longrightarrow> \<guillemotleft>\<t>[a] : a \<rightarrow> \<one>\<guillemotright>"
   and trm_eqI: "\<lbrakk> ide a; \<guillemotleft>f : a \<rightarrow> \<one>\<guillemotright> \<rbrakk> \<Longrightarrow> f = \<t>[a]"
   begin
 
     lemma trm_simps:
     assumes "ide a"
     shows "arr \<t>[a]" and "dom \<t>[a] = a" and "cod \<t>[a] = \<one>"
       using assms trm_in_hom by auto
 
     lemma trm_one:
     shows "\<t>[\<one>] = \<one>"
     using ide_one trm_in_hom trm_eqI ide_in_hom by auto
 
     lemma terminal_one:
     shows "terminal \<one>"
       using ide_one trm_in_hom trm_eqI terminal_def by metis
 
     lemma trm_naturality:
     assumes "arr f"
     shows "\<t>[cod f] \<cdot> f = \<t>[dom f]"
       using assms trm_eqI
       by (metis comp_in_homI' ide_cod ide_dom in_homE trm_in_hom)
 
     proposition is_category_with_terminal_object:
     shows "category_with_terminal_object C"
       apply unfold_locales
       using terminal_one by auto
 
   end
 
   context category_with_terminal_object
   begin
 
     definition some_terminal ("\<one>")
     where "some_terminal \<equiv> SOME t. terminal t"
 
     definition "trm" ("\<t>[_]")
     where "\<t>[f] \<equiv> if arr f then THE t. \<guillemotleft>t : dom f \<rightarrow> \<one>\<guillemotright> else null"
 
     lemma terminal_some_terminal [intro]:
     shows "terminal \<one>"
       using some_terminal_def has_terminal someI_ex [of "\<lambda>t. terminal t"] by presburger
 
     lemma ide_some_terminal:
     shows "ide \<one>"
       using terminal_def by blast
 
     lemma trm_in_hom [intro]:
     assumes "arr f"
     shows "\<guillemotleft>\<t>[f] : dom f \<rightarrow> \<one>\<guillemotright>"
     proof -
       have "ide (dom f)" using assms by fastforce
       hence "\<exists>!t. \<guillemotleft>t : dom f \<rightarrow> \<one>\<guillemotright>"
         using assms trm_def terminal_def terminal_some_terminal by simp
       thus ?thesis
         using assms trm_def [of f] theI' [of "\<lambda>t. \<guillemotleft>t : dom f \<rightarrow> \<one>\<guillemotright>"] by auto
     qed
 
     lemma trm_simps [simp]:
     assumes "arr f"
     shows "arr \<t>[f]" and "dom \<t>[f] = dom f" and "cod \<t>[f] = \<one>"
       using assms trm_in_hom by auto
 
     lemma trm_eqI:
     assumes "\<guillemotleft>t : dom f \<rightarrow> \<one>\<guillemotright>"
     shows "t = \<t>[f]"
     proof -
       have "ide (dom f)" using assms
         by (metis ide_dom in_homE)
       hence "\<exists>!t. \<guillemotleft>t : dom f \<rightarrow> \<one>\<guillemotright>"
         using terminal_def [of \<one>] terminal_some_terminal by auto
       moreover have "\<guillemotleft>t : dom f \<rightarrow> \<one>\<guillemotright>"
         using assms by simp
       ultimately show ?thesis
         using assms trm_def the1_equality [of "\<lambda>t. \<guillemotleft>t : dom f \<rightarrow> \<one>\<guillemotright>" t]
               \<open>ide (dom f)\<close> arr_dom_iff_arr
         by fastforce
     qed
 
     sublocale elementary_category_with_terminal_object C \<one> trm
       using ide_some_terminal trm_eqI
       by unfold_locales auto
 
     proposition extends_to_elementary_category_with_terminal_object:
       shows "elementary_category_with_terminal_object C \<one> trm"
       ..
 
   end
 
   section "Cartesian Category"
 
   locale cartesian_category =
     category_with_binary_products +
     category_with_terminal_object
 
   locale elementary_cartesian_category =
     elementary_category_with_binary_products +
     elementary_category_with_terminal_object
   begin
 
     proposition is_cartesian_category:
     shows "cartesian_category C"
       using cartesian_category.intro is_category_with_binary_products
             is_category_with_terminal_object
       by auto
 
   end
 
   context cartesian_category
   begin
 
     proposition extends_to_elementary_cartesian_category:
     shows "elementary_cartesian_category C pr0 pr1 \<one> trm"
       by (simp add: elementary_cartesian_category_def
           elementary_category_with_terminal_object_axioms
           extends_to_elementary_category_with_binary_products)
 
     sublocale elementary_cartesian_category C pr0 pr1 \<one> trm
       using extends_to_elementary_cartesian_category by simp
 
   end
 
   text \<open>
     Here we prove some facts that will later allow us to show that an elementary cartesian
     category is a monoidal category.
   \<close>
 
   context elementary_cartesian_category
   begin
 
     abbreviation \<iota>
     where "\<iota> \<equiv> \<pp>\<^sub>0[\<one>, \<one>]"
 
     lemma pr_coincidence:
     shows "\<iota> = \<pp>\<^sub>1[\<one>, \<one>]"
       using ide_one
       by (simp add: terminal_arr_unique terminal_one)
 
     lemma \<iota>_is_terminal_arr:
     shows "terminal_arr \<iota>"
       using ide_one
       by (simp add: terminal_one)
 
     lemma inverse_arrows_\<iota>:
     shows "inverse_arrows \<iota> \<langle>\<one>, \<one>\<rangle>"
       using ide_one
       by (metis (no_types, lifting) dup_is_natural_transformation \<iota>_is_terminal_arr cod_pr0
           comp_cod_arr pr_dup(1) ide_dom inverse_arrows_def map_simp
           natural_transformation.is_natural_2 pr_simps(2) pr1_in_hom' trm_eqI trm_naturality
           trm_one tuple_pr)
 
     lemma \<iota>_is_iso:
     shows "iso \<iota>"
       using inverse_arrows_\<iota> by auto
 
     lemma trm_tensor:
     assumes "ide a" and "ide b"
     shows "\<t>[a \<otimes> b] = \<iota> \<cdot> (\<t>[a] \<otimes> \<t>[b])"
     proof -
       have "\<t>[a \<otimes> b] = \<t>[a] \<cdot> \<pp>\<^sub>1[a, b]"
         by (metis assms(1-2) cod_pr1 pr_simps(4-6) trm_naturality)
       moreover have "\<guillemotleft>\<t>[b] : b \<rightarrow> \<one>\<guillemotright>"
         using assms(2) trm_in_hom by blast
       ultimately show ?thesis
         using assms(1) pr_coincidence trm_in_hom by fastforce
     qed
 
     abbreviation runit ("\<r>[_]")
     where "\<r>[a] \<equiv> \<pp>\<^sub>1[a, \<one>]"
 
     abbreviation runit' ("\<r>\<^sup>-\<^sup>1[_]")
     where "\<r>\<^sup>-\<^sup>1[a] \<equiv> \<langle>a, \<t>[a]\<rangle>"
 
     abbreviation lunit ("\<l>[_]")
     where "\<l>[a] \<equiv> \<pp>\<^sub>0[\<one>, a]"
 
     abbreviation lunit' ("\<l>\<^sup>-\<^sup>1[_]")
     where "\<l>\<^sup>-\<^sup>1[a] \<equiv> \<langle>\<t>[a], a\<rangle>"
 
     lemma runit_in_hom:
     assumes "ide a"
     shows "\<guillemotleft>\<r>[a] : a \<otimes> \<one> \<rightarrow> a\<guillemotright>"
      using assms ide_one by simp
 
     lemma runit'_in_hom:
     assumes "ide a"
     shows "\<guillemotleft>\<r>\<^sup>-\<^sup>1[a] : a \<rightarrow> a \<otimes> \<one>\<guillemotright>"
       using assms ide_in_hom trm_in_hom by blast
 
     lemma lunit_in_hom:
     assumes "ide a"
     shows "\<guillemotleft>\<l>[a] : \<one> \<otimes> a \<rightarrow> a\<guillemotright>"
      using assms ide_one by simp
 
     lemma lunit'_in_hom:
     assumes "ide a"
     shows "\<guillemotleft>\<l>\<^sup>-\<^sup>1[a] : a \<rightarrow> \<one> \<otimes> a\<guillemotright>"
       using assms ide_in_hom trm_in_hom by blast
 
     lemma runit_naturality:
     assumes "ide a"
     shows "\<r>[cod a] \<cdot> (a \<otimes> \<one>) = a \<cdot> \<r>[dom a]"
       using assms pr_naturality(2) ide_char ide_one by blast
 
     lemma inverse_arrows_runit:
     assumes "ide a"
     shows "inverse_arrows \<r>[a] \<r>\<^sup>-\<^sup>1[a]"
     proof
       show "ide (\<r>[a] \<cdot> \<r>\<^sup>-\<^sup>1[a])"
       proof -
         have "\<r>[a] \<cdot> \<r>\<^sup>-\<^sup>1[a] = a"
           using assms
           by (metis in_homE ide_char pr_tuple(1) trm_in_hom)
         thus ?thesis
           using assms by presburger
       qed
       show "ide (\<r>\<^sup>-\<^sup>1[a] \<cdot> \<r>[a])"
       proof -
         have "ide (a \<otimes> \<one>)"
           using assms ide_one by blast
         moreover have "\<r>\<^sup>-\<^sup>1[a] \<cdot> \<r>[a] = a \<otimes> \<one>"
         proof (intro pr_joint_monic [of a \<one> "\<r>\<^sup>-\<^sup>1[a] \<cdot> \<r>[a]" "a \<otimes> \<one>"])
           show "seq \<pp>\<^sub>0[a, \<one>] (\<r>\<^sup>-\<^sup>1[a] \<cdot> \<r>[a])"
             using assms ide_one runit'_in_hom [of a]
             by (intro seqI) auto
           show "\<pp>\<^sub>0[a, \<one>] \<cdot> \<r>\<^sup>-\<^sup>1[a] \<cdot> \<r>[a] = \<pp>\<^sub>0[a, \<one>] \<cdot> (a \<otimes> \<one>)"
           proof -
             have "\<pp>\<^sub>0[a, \<one>] \<cdot> \<r>\<^sup>-\<^sup>1[a] \<cdot> \<r>[a] = (\<pp>\<^sub>0[a, \<one>] \<cdot> \<r>\<^sup>-\<^sup>1[a]) \<cdot> \<r>[a]"
               using comp_assoc by simp
             also have "... = \<t>[a] \<cdot> \<r>[a]"
               using assms ide_one
               by (metis in_homE pr_tuple(2) ide_char trm_in_hom)
             also have "... = \<t>[a \<otimes> \<one>]"
               using assms ide_one trm_naturality [of "\<r>[a]"] by simp
             also have "... = \<pp>\<^sub>0[a, \<one>] \<cdot> (a \<otimes> \<one>)"
               using assms comp_arr_dom ide_one trm_naturality trm_one by fastforce
             finally show ?thesis by blast
           qed
           show "\<pp>\<^sub>1[a, \<one>] \<cdot> \<r>\<^sup>-\<^sup>1[a] \<cdot> \<r>[a] = \<pp>\<^sub>1[a, \<one>] \<cdot> (a \<otimes> \<one>)"
             using assms
             by (metis \<open>ide (\<r>[a] \<cdot> \<r>\<^sup>-\<^sup>1[a])\<close> cod_comp cod_pr1 dom_comp ide_compE ide_one
                 comp_assoc runit_naturality)
         qed
         ultimately show ?thesis by simp
       qed
     qed
 
     lemma lunit_naturality:
     assumes "arr f"
     shows "C \<l>[cod f] (\<one> \<otimes> f) = C f \<l>[dom f]"
       using assms pr_naturality(1) ide_char ide_one by blast
 
     lemma inverse_arrows_lunit:
     assumes "ide a"
     shows "inverse_arrows \<l>[a] \<l>\<^sup>-\<^sup>1[a]"
     proof
       show "ide (C \<l>[a] \<l>\<^sup>-\<^sup>1[a])"
       proof -
         have "C \<l>[a] \<l>\<^sup>-\<^sup>1[a] = a"
           using assms
           by (metis ide_char in_homE pr_tuple(2) trm_in_hom)
         thus ?thesis
           using assms by simp
       qed
       show "ide (\<l>\<^sup>-\<^sup>1[a] \<cdot> \<l>[a])"
       proof -
         have "\<l>\<^sup>-\<^sup>1[a] \<cdot> \<l>[a] = \<one> \<otimes> a"
         proof (intro pr_joint_monic [of \<one> a "\<l>\<^sup>-\<^sup>1[a] \<cdot> \<l>[a]" "\<one> \<otimes> a"])
           show "seq \<l>[a] (\<l>\<^sup>-\<^sup>1[a] \<cdot> \<l>[a])"
             using assms \<open>ide (\<l>[a] \<cdot> \<l>\<^sup>-\<^sup>1[a])\<close> by blast
           show "\<l>[a] \<cdot> \<l>\<^sup>-\<^sup>1[a] \<cdot> \<l>[a] = \<l>[a] \<cdot> (\<one> \<otimes> a)"
             using assms
             by (metis \<open>ide (\<l>[a] \<cdot> \<l>\<^sup>-\<^sup>1[a])\<close> cod_comp cod_pr0 dom_cod ide_compE ide_one
                 comp_assoc lunit_naturality)
           show "\<pp>\<^sub>1[\<one>, a] \<cdot> \<l>\<^sup>-\<^sup>1[a] \<cdot> \<l>[a] = \<pp>\<^sub>1[\<one>, a] \<cdot> (\<one> \<otimes> a)"
           proof -
             have "\<pp>\<^sub>1[\<one>, a] \<cdot> \<l>\<^sup>-\<^sup>1[a] \<cdot> \<l>[a] = (\<pp>\<^sub>1[\<one>, a] \<cdot> \<l>\<^sup>-\<^sup>1[a]) \<cdot> \<l>[a]"
               using comp_assoc by simp
             also have "... = \<t>[a] \<cdot> \<l>[a]"
               using assms ide_one
               by (metis pr_tuple(1) ide_char in_homE trm_in_hom)
             also have "... = \<t>[\<one> \<otimes> a]"
               using assms ide_one trm_naturality [of "\<l>[a]"] by simp
             also have "... = \<pp>\<^sub>1[\<one>, a] \<cdot> (\<one> \<otimes> a)"
               using assms comp_arr_dom ide_one trm_naturality trm_one by fastforce
             finally show ?thesis by simp
           qed
         qed
         moreover have "ide (\<one> \<otimes> a)"
           using assms ide_one by simp
         finally show ?thesis by blast
       qed
     qed
 
     lemma comp_lunit_term_dup:
     assumes "ide a"
     shows "\<l>[a] \<cdot> (\<t>[a] \<otimes> a) \<cdot> \<d>[a] = a"
     proof -
       have "\<guillemotleft>\<t>[a] : a \<rightarrow> \<one>\<guillemotright>"
         using assms trm_in_hom by blast
       hence "\<l>[a] \<cdot> (\<t>[a] \<otimes> a) = a \<cdot> \<pp>\<^sub>0[a, a]"
         by (metis assms pr_naturality(1) ide_char in_homE)
       thus ?thesis
         by (metis (no_types) assms comp_assoc comp_ide_self pr_dup(1))
     qed
 
     lemma comp_runit_term_dup:
     assumes "ide a"
     shows "\<r>[a] \<cdot> (a \<otimes> \<t>[a]) \<cdot> \<d>[a] = a"
     proof -
       have "\<guillemotleft>\<t>[a] : a \<rightarrow> \<one>\<guillemotright>"
         using assms trm_in_hom by blast
       hence "\<r>[a] \<cdot> (a \<otimes> \<t>[a]) = a \<cdot> \<pp>\<^sub>1[a, a]"
         using assms by auto
       thus ?thesis
         using assms
         by (metis comp_ide_arr pr_dup(2) ide_char comp_assoc seqI)
     qed
 
     lemma comp_proj_assoc:
     assumes "ide a0" and "ide a1" and "ide a2"
     shows "\<pp>\<^sub>1[a0, a1 \<otimes> a2] \<cdot> \<a>[a0, a1, a2] = \<pp>\<^sub>1[a0, a1] \<cdot> \<pp>\<^sub>1[a0 \<otimes> a1, a2]"
     and "\<pp>\<^sub>0[a0, a1 \<otimes> a2] \<cdot> \<a>[a0, a1, a2] = \<langle>\<pp>\<^sub>0[a0, a1] \<cdot> \<pp>\<^sub>1[a0 \<otimes> a1, a2], \<pp>\<^sub>0[a0 \<otimes> a1, a2]\<rangle>"
       using assms assoc_def by auto
 
     lemma dup_coassoc:
     assumes "ide a"
     shows "\<a>[a, a, a] \<cdot> (\<d>[a] \<otimes> a) \<cdot> \<d>[a] = (a \<otimes> \<d>[a]) \<cdot> \<d>[a]"
     proof (intro pr_joint_monic
                    [of a "a \<otimes> a" "\<a>[a, a, a] \<cdot> (\<d>[a] \<otimes> a) \<cdot> \<d>[a]" "(a \<otimes> \<d>[a]) \<cdot> \<d>[a]"])
       show "seq \<pp>\<^sub>0[a, a \<otimes> a] (\<a>[a, a, a] \<cdot> (\<d>[a] \<otimes> a) \<cdot> \<d>[a])"
         using assms by simp
       show "\<pp>\<^sub>0[a, a \<otimes> a] \<cdot> \<a>[a, a, a] \<cdot> (\<d>[a] \<otimes> a) \<cdot> \<d>[a] = \<pp>\<^sub>0[a, a \<otimes> a] \<cdot> (a \<otimes> \<d>[a]) \<cdot> \<d>[a]"
       proof -
         have "\<pp>\<^sub>0[a, a \<otimes> a] \<cdot> \<a>[a, a, a] \<cdot> (\<d>[a] \<otimes> a) \<cdot> \<d>[a] =
               ((\<pp>\<^sub>0[a, a \<otimes> a] \<cdot> \<a>[a, a, a]) \<cdot> (\<d>[a] \<otimes> a)) \<cdot> \<d>[a]"
           using comp_assoc by simp
         also have "... = \<langle>((\<pp>\<^sub>0[a, a] \<cdot> \<pp>\<^sub>1[a \<otimes> a, a]) \<cdot> (\<d>[a] \<otimes> a)) \<cdot> \<d>[a], (a \<cdot> \<pp>\<^sub>0[a, a]) \<cdot> \<d>[a]\<rangle>"
           using assms assoc_def by simp
         also have "... = \<d>[a]"
           using assms comp_assoc by simp
         also have "... = (\<pp>\<^sub>0[a, a \<otimes> a] \<cdot> (a \<otimes> \<d>[a])) \<cdot> \<d>[a]"
           using assms assoc_def comp_assoc by simp
         also have "... = \<pp>\<^sub>0[a, a \<otimes> a] \<cdot> (a \<otimes> \<d>[a]) \<cdot> \<d>[a]"
           using comp_assoc by simp
         finally show ?thesis by blast
       qed
       show "\<pp>\<^sub>1[a, a \<otimes> a] \<cdot> \<a>[a, a, a] \<cdot> (\<d>[a] \<otimes> a) \<cdot> \<d>[a] = \<pp>\<^sub>1[a, a \<otimes> a] \<cdot> (a \<otimes> \<d>[a]) \<cdot> \<d>[a]"
       proof -
         have "\<pp>\<^sub>1[a, a \<otimes> a] \<cdot> \<a>[a, a, a] \<cdot> (\<d>[a] \<otimes> a) \<cdot> \<d>[a] =
               ((\<pp>\<^sub>1[a, a \<otimes> a] \<cdot> \<a>[a, a, a]) \<cdot> (\<d>[a] \<otimes> a)) \<cdot> \<d>[a]"
           using comp_assoc by simp
         also have "... = ((\<pp>\<^sub>1[a, a] \<cdot> \<pp>\<^sub>1[a \<otimes> a, a]) \<cdot> (\<d>[a] \<otimes> a)) \<cdot> \<d>[a]"
           using assms assoc_def by simp
         also have "... = a"
           using assms comp_assoc by simp
         also have "... = (a \<cdot> \<pp>\<^sub>1[a, a]) \<cdot> \<d>[a]"
           using assms comp_assoc by simp
         also have "... = (\<pp>\<^sub>1[a, a \<otimes> a] \<cdot> (a \<otimes> \<d>[a])) \<cdot> \<d>[a]"
           using assms by simp
         also have "... = \<pp>\<^sub>1[a, a \<otimes> a] \<cdot> (a \<otimes> \<d>[a]) \<cdot> \<d>[a]"
           using comp_assoc by simp
         finally show ?thesis by blast
       qed
     qed
 
     lemma comp_assoc_tuple:
     assumes "\<guillemotleft>f0 : a \<rightarrow> b0\<guillemotright>" and "\<guillemotleft>f1 : a \<rightarrow> b1\<guillemotright>" and "\<guillemotleft>f2 : a \<rightarrow> b2\<guillemotright>"
     shows "\<a>[b0, b1, b2] \<cdot> \<langle>\<langle>f0, f1\<rangle>, f2\<rangle> = \<langle>f0, \<langle>f1, f2\<rangle>\<rangle>"
     and "\<a>\<^sup>-\<^sup>1[b0, b1, b2] \<cdot> \<langle>f0, \<langle>f1, f2\<rangle>\<rangle> = \<langle>\<langle>f0, f1\<rangle>, f2\<rangle>"
       using assms assoc_def assoc'_def comp_assoc by fastforce+
 
     lemma dup_tensor:
     assumes "ide a" and "ide b"
     shows "\<d>[a \<otimes> b] = \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> (a \<otimes> \<a>[b, a, b]) \<cdot> (a \<otimes> \<sigma> (a, b) \<otimes> b) \<cdot>
                         (a \<otimes> \<a>\<^sup>-\<^sup>1[a, b, b]) \<cdot> \<a>[a, a, b \<otimes> b] \<cdot> (\<d>[a] \<otimes> \<d>[b])"
     proof (intro pr_joint_monic [of "a \<otimes> b" "a \<otimes> b" "\<d>[a \<otimes> b]"])
       show "seq \<pp>\<^sub>0[a \<otimes> b, a \<otimes> b] (\<d>[a \<otimes> b])"
         using assms by simp
       have 1: "\<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> (a \<otimes> \<a>[b, a, b]) \<cdot> (a \<otimes> \<sigma> (a, b) \<otimes> b) \<cdot>
                  (a \<otimes> \<a>\<^sup>-\<^sup>1[a, b, b]) \<cdot> \<a>[a, a, b \<otimes> b] \<cdot> (\<d>[a] \<otimes> \<d>[b]) =
                \<langle>a \<otimes> b, a \<otimes> b\<rangle>"
       proof -
         have "\<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> (a \<otimes> \<a>[b, a, b]) \<cdot> (a \<otimes> \<sigma> (a, b) \<otimes> b) \<cdot>
               (a \<otimes> \<a>\<^sup>-\<^sup>1[a, b, b]) \<cdot> \<a>[a, a, b \<otimes> b] \<cdot> (\<d>[a] \<otimes> \<d>[b])
                 = \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> (a \<otimes> \<a>[b, a, b]) \<cdot> (a \<otimes> \<sigma> (a, b) \<otimes> b) \<cdot>
                   (a \<otimes> \<a>\<^sup>-\<^sup>1[a, b, b]) \<cdot> \<langle>\<pp>\<^sub>1[a, b], \<langle>\<pp>\<^sub>1[a, b], \<d>[b] \<cdot> \<pp>\<^sub>0[a, b]\<rangle>\<rangle>"
         proof -
           have "\<a>[a, a, b \<otimes> b] \<cdot> (\<d>[a] \<otimes> \<d>[b]) = \<langle>\<pp>\<^sub>1[a, b], \<langle>\<pp>\<^sub>1[a, b], \<d>[b] \<cdot> \<pp>\<^sub>0[a, b]\<rangle>\<rangle>"
             using assms assoc_def comp_assoc pr_naturality comp_cod_arr by simp
           thus ?thesis by presburger
         qed
         also have "... = \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot>
         \<langle>a \<cdot> a \<cdot> a \<cdot> \<pp>\<^sub>1[a, b], \<a>[b, a, b] \<cdot> (\<s>[a, b] \<cdot> (a \<otimes> b) \<otimes> b) \<cdot>
                                \<a>\<^sup>-\<^sup>1[a, b, b] \<cdot> \<langle>\<pp>\<^sub>1[a, b], \<d>[b \<cdot> \<pp>\<^sub>0[a, b]]\<rangle>\<rangle>"
           using assms prod_tuple by simp
         also have "... = \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot>
         \<langle>\<pp>\<^sub>1[a, b], \<a>[b, a, b] \<cdot> (\<s>[a, b] \<otimes> b) \<cdot> \<a>\<^sup>-\<^sup>1[a, b, b] \<cdot> \<langle>\<pp>\<^sub>1[a, b], \<d>[\<pp>\<^sub>0[a, b]]\<rangle>\<rangle>"
         proof -
           have "a \<cdot> a \<cdot> a \<cdot> \<pp>\<^sub>1[a, b] = \<pp>\<^sub>1[a, b]"
             using assms comp_cod_arr by simp
           moreover have "b \<cdot> \<pp>\<^sub>0[a, b] = \<pp>\<^sub>0[a, b]"
             using assms comp_cod_arr by simp
           moreover have "\<s>[a, b] \<cdot> (a \<otimes> b) \<otimes> b = \<s>[a, b] \<otimes> b"
             using assms comp_arr_dom by simp
           ultimately show ?thesis by simp
         qed
         also have "... = \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> \<langle>\<pp>\<^sub>1[a, b], \<a>[b, a, b] \<cdot> (\<s>[a, b] \<otimes> b) \<cdot>
                            \<langle>\<langle>\<pp>\<^sub>1[a, b], \<pp>\<^sub>0[a, b]\<rangle>, \<pp>\<^sub>0[a, b]\<rangle>\<rangle>"
         proof -
           have "\<a>\<^sup>-\<^sup>1[a, b, b] \<cdot> \<langle>\<pp>\<^sub>1[a, b], \<d>[\<pp>\<^sub>0[a, b]]\<rangle> = \<langle>\<langle>\<pp>\<^sub>1[a, b], \<pp>\<^sub>0[a, b]\<rangle>, \<pp>\<^sub>0[a, b]\<rangle>"
             using assms comp_assoc_tuple(2) by blast
           thus ?thesis by simp
         qed
         also have "... = \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> \<langle>\<pp>\<^sub>1[a, b], \<a>[b, a, b] \<cdot> \<langle>\<s>[a, b], \<pp>\<^sub>0[a, b]\<rangle>\<rangle>"
           using assms prod_tuple comp_arr_dom comp_cod_arr by simp
         also have "... = \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> \<langle>\<pp>\<^sub>1[a, b], \<langle>\<pp>\<^sub>0[a, b], \<langle>\<pp>\<^sub>1[a, b], \<pp>\<^sub>0[a, b]\<rangle>\<rangle>\<rangle>"
           using assms comp_assoc_tuple(1)
           by (metis sym_def pr_in_hom)
         also have "... = \<langle>\<langle>\<pp>\<^sub>1[a, b], \<pp>\<^sub>0[a, b]\<rangle>, \<langle>\<pp>\<^sub>1[a, b], \<pp>\<^sub>0[a, b]\<rangle>\<rangle>"
           using assms comp_assoc_tuple(2) by force
         also have "... = \<d>[a \<otimes> b]"
           using assms by simp
         finally show ?thesis by simp
       qed
       show "\<pp>\<^sub>0[a \<otimes> b, a \<otimes> b] \<cdot> \<d>[a \<otimes> b]
               = \<pp>\<^sub>0[a \<otimes> b, a \<otimes> b] \<cdot>
                 \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> (a \<otimes> \<a>[b, a, b]) \<cdot> (a \<otimes> \<sigma> (a, b) \<otimes> b) \<cdot>
                 (a \<otimes> \<a>\<^sup>-\<^sup>1[a, b, b]) \<cdot> \<a>[a, a, b \<otimes> b] \<cdot> (\<d>[a] \<otimes> \<d>[b])"
         using assms 1 by force
       show "\<pp>\<^sub>1[a \<otimes> b, a \<otimes> b] \<cdot> \<d>[a \<otimes> b]
               = \<pp>\<^sub>1[a \<otimes> b, a \<otimes> b] \<cdot>
                 \<a>\<^sup>-\<^sup>1[a, b, a \<otimes> b] \<cdot> (a \<otimes> \<a>[b, a, b]) \<cdot> (a \<otimes> \<sigma> (a, b) \<otimes> b) \<cdot>
                 (a \<otimes> \<a>\<^sup>-\<^sup>1[a, b, b]) \<cdot> \<a>[a, a, b \<otimes> b] \<cdot> (\<d>[a] \<otimes> \<d>[b])"
         using assms 1 by force
     qed
 
     (* TODO: Not sure if the remaining facts are useful. *)
 
     lemma \<iota>_eq_trm:
     shows "\<iota> = \<t>[\<one> \<otimes> \<one>]"
     proof (intro terminal_arr_unique)
       show "par \<iota> \<t>[\<one> \<otimes> \<one>]"
         by (simp add: ide_one trm_one trm_tensor)
       show "terminal_arr \<t>[\<one> \<otimes> \<one>]"
         using ide_one \<iota>_is_terminal_arr \<open>par \<iota> \<t>[\<one> \<otimes> \<one>]\<close> by auto
       show "terminal_arr \<iota>"
         using \<iota>_is_terminal_arr by blast
     qed
 
     lemma terminal_tensor_one_one:
     shows "terminal (\<one> \<otimes> \<one>)"
     proof
       show "ide (\<one> \<otimes> \<one>)"
         using ide_one by simp
       show "\<And>a. ide a \<Longrightarrow> \<exists>!f. \<guillemotleft>f : a \<rightarrow> \<one> \<otimes> \<one>\<guillemotright>"
       proof -
         fix a
         assume a: "ide a"
         show "\<exists>!f. \<guillemotleft>f : a \<rightarrow> \<one> \<otimes> \<one>\<guillemotright>"
         proof
           show "\<guillemotleft>inv \<iota> \<cdot> \<t>[a] : a \<rightarrow> \<one> \<otimes> \<one>\<guillemotright>"
             using a ide_one inverse_arrows_\<iota> inverse_unique trm_in_hom by fastforce
           show "\<And>f. \<guillemotleft>f : a \<rightarrow> \<one> \<otimes> \<one>\<guillemotright> \<Longrightarrow> f = inv \<iota> \<cdot> \<t>[a]"
           proof -
             fix f
             assume f: "\<guillemotleft>f : a \<rightarrow> \<one> \<otimes> \<one>\<guillemotright>"
             have "\<iota> \<cdot> f = \<t>[a]"
             proof (intro terminal_arr_unique)
               show "par (\<iota> \<cdot> f) \<t>[a]"
                 using a f
                 by (metis \<iota>_is_iso \<iota>_is_terminal_arr \<open>\<guillemotleft>inv \<iota> \<cdot> \<t>[a] : a \<rightarrow> \<one> \<otimes> \<one>\<guillemotright>\<close>
                     cod_comp dom_comp dom_inv ide_one in_homE pr_simps(2-3) seqE seqI)
               show "terminal_arr (\<iota> \<cdot> f)"
                 using a f \<iota>_is_terminal_arr cod_comp by force
               show "terminal_arr \<t>[a]"
                 using a \<open>par (\<iota> \<cdot> f) \<t>[a]\<close> \<open>terminal_arr (\<iota> \<cdot> f)\<close> by auto
             qed
             thus "f = inv \<iota> \<cdot> \<t>[a]"
               using a f \<iota>_is_iso invert_side_of_triangle(1)
                     \<open>\<guillemotleft>inv \<iota> \<cdot> \<t>[a] : a \<rightarrow> \<one> \<otimes> \<one>\<guillemotright>\<close>
               by blast
           qed
         qed
       qed
     qed
 
   end
 
   section "Category with Finite Products"
 
   text \<open>
     In this last section, we show that the notion ``cartesian category'', which we defined
     to be a category with binary products and terminal object, coincides with the notion
     ``category with finite products''.  Due to the inability to quantify over types in HOL,
     we content ourselves with defining the latter notion as "has \<open>I\<close>-indexed products
     for every finite set \<open>I\<close> of natural numbers."  We can transfer this property to finite
     sets at other types using the fact that products are preserved under bijections of
     the index sets.
   \<close>
 
   locale category_with_finite_products =
     category C
   for C :: "'c comp" +
   assumes has_finite_products: "finite (I :: nat set) \<Longrightarrow> has_products I"
   begin
 
     lemma has_finite_products':
     assumes "I \<noteq> UNIV"
     shows "finite I \<Longrightarrow> has_products I"
     proof -
       assume I: "finite I"
       obtain n \<phi> where \<phi>: "bij_betw \<phi> {k. k < (n :: nat)} I"
         using I finite_imp_nat_seg_image_inj_on inj_on_imp_bij_betw by fastforce
       show "has_products I"
         using assms(1) \<phi> has_finite_products has_products_preserved_by_bijection
               category_with_finite_products.has_finite_products
         by blast
     qed
 
   end
 
   lemma (in category) has_binary_products_if:
   assumes "has_products ({0, 1} :: nat set)"
   shows "has_binary_products"
   proof (unfold has_binary_products_def)
     show "\<forall>a0 a1. ide a0 \<and> ide a1 \<longrightarrow> (\<exists>p0 p1. has_as_binary_product a0 a1 p0 p1)"
     proof (intro allI impI)
       fix a0 a1
       assume 1: "ide a0 \<and> ide a1"
       show "\<exists>p0 p1. has_as_binary_product a0 a1 p0 p1"
       proof -
         interpret J: binary_product_shape
           by unfold_locales
         interpret D: binary_product_diagram C a0 a1
           using 1 by unfold_locales auto
         interpret discrete_diagram J.comp C D.map
           using J.is_discrete
           by unfold_locales auto
         show "\<exists>p0 p1. has_as_binary_product a0 a1 p0 p1"
         proof (unfold has_as_binary_product_def)
           text \<open>
             Here we have to work around the fact that \<open>has_finite_products\<close> is defined
             in terms of @{typ "nat set"}, whereas \<open>has_as_binary_product\<close> is defined
             in terms of \<open>J.arr set\<close>.
           \<close>
           let ?\<phi> = "(\<lambda>x :: nat. if x = 0 then J.FF else J.TT)"
           let ?\<psi> = "\<lambda>j. if j = J.FF then 0 else 1"
           have "bij_betw ?\<phi> ({0, 1} :: nat set) {J.FF, J.TT}"
             using bij_betwI [of ?\<phi> "{0, 1} :: nat set" "{J.FF, J.TT}" ?\<psi>] by fastforce
           hence "has_products {J.FF, J.TT}"
             using assms has_products_def [of "{J.FF, J.TT}"]
                   has_products_preserved_by_bijection
                     [of "{0, 1} :: nat set" ?\<phi> "{J.FF, J.TT}"]
             by blast
           hence "\<exists>a. has_as_product J.comp D.map a"
             using has_products_def [of "{J.FF, J.TT}"]
                   discrete_diagram_axioms J.arr_char
             by blast
           hence "\<exists>a \<pi>. product_cone J.comp C D.map a \<pi>"
             using has_as_product_def by blast
           hence 2: "\<exists>a \<pi>. D.limit_cone a \<pi>"
             unfolding product_cone_def by simp
           obtain a \<pi> where \<pi>: "D.limit_cone a \<pi>"
             using 2 by auto
           interpret \<pi>: limit_cone J.comp C D.map a \<pi>
             using \<pi> by auto
           have "\<pi> = D.mkCone (\<pi> J.FF) (\<pi> J.TT)"
           proof -
             have "\<And>a. J.ide a \<Longrightarrow> \<pi> a = D.mkCone (\<pi> J.FF) (\<pi> J.TT) a"
               using D.mkCone_def J.ide_char by auto
             moreover have "a = dom (\<pi> J.FF)"
               by simp
             moreover have "D.cone a (D.mkCone (\<pi> (J.MkIde False)) (\<pi> (J.MkIde True)))"
               using 1 D.cone_mkCone [of "\<pi> J.FF" "\<pi> J.TT"] by auto
             ultimately show ?thesis
               using D.mkCone_def \<pi>.natural_transformation_axioms
                     D.cone_mkCone [of "\<pi> J.FF" "\<pi> J.TT"]
                     NaturalTransformation.eqI
                       [of "J.comp" C \<pi>.A.map "D.map" \<pi> "D.mkCone (\<pi> J.FF) (\<pi> J.TT)"]
                     cone_def [of J.comp C D.map a "D.mkCone (\<pi> J.FF) (\<pi> J.TT)"] J.ide_char
               by blast
           qed
           hence "D.limit_cone (dom (\<pi> J.FF)) (D.mkCone (\<pi> J.FF) (\<pi> J.TT))"
             using \<pi>.limit_cone_axioms by simp
           thus "\<exists>p0 p1. ide a0 \<and> ide a1 \<and> D.has_as_binary_product p0 p1"
             using 1 by blast
         qed
       qed
     qed
   qed
 
   sublocale category_with_finite_products \<subseteq> category_with_binary_products C
     using has_binary_products_if has_finite_products
     by (unfold_locales, unfold has_binary_products_def) simp
 
   proposition (in category_with_finite_products) is_category_with_binary_products:
   shows "category_with_binary_products C"
     ..
 
   sublocale category_with_finite_products \<subseteq> category_with_terminal_object C
   proof
     interpret J: discrete_category "{} :: nat set"
       by unfold_locales auto
     interpret D: empty_diagram J.comp C "\<lambda>j. null"
       by unfold_locales auto
     interpret D: discrete_diagram J.comp C "\<lambda>j. null"
       using J.is_discrete by unfold_locales auto
     have "\<And>a. D.has_as_limit a \<longleftrightarrow> has_as_product J.comp (\<lambda>j. null) a"
       using product_cone_def J.category_axioms category_axioms D.discrete_diagram_axioms
             has_as_product_def product_cone_def
       by metis
     moreover have "\<exists>a. has_as_product J.comp (\<lambda>j. null) a"
       using has_finite_products [of "{} :: nat set"] has_products_def [of "{} :: nat set"]
             D.discrete_diagram_axioms
       by blast
     ultimately have "\<exists>a. D.has_as_limit a" by blast
     thus "\<exists>a. terminal a" using D.has_as_limit_iff_terminal by blast
   qed
 
   proposition (in category_with_finite_products) is_category_with_terminal_object:
   shows "category_with_terminal_object C"
     ..
 
   sublocale category_with_finite_products \<subseteq> cartesian_category ..
 
   proposition (in category_with_finite_products) is_cartesian_category:
   shows "cartesian_category C"
     ..
 
   context category
   begin
 
     lemma binary_product_of_products_is_product:
     assumes "has_as_product J0 D0 a0" and "has_as_product J1 D1 a1"
     and "has_as_binary_product a0 a1 p0 p1"
     and "Collect (partial_magma.arr J0) \<inter> Collect (partial_magma.arr J1) = {}"
     and "partial_magma.null J0 = partial_magma.null J1"
     shows "has_as_product
              (discrete_category.comp
                 (Collect (partial_magma.arr J0) \<union> Collect (partial_magma.arr J1))
                 (partial_magma.null J0))
              (\<lambda>i. if i \<in> Collect (partial_magma.arr J0) then D0 i
                   else if i \<in> Collect (partial_magma.arr J1) then D1 i
                   else null)
              (dom p0)"
     proof -
       obtain \<pi>0 where \<pi>0: "product_cone J0 (\<cdot>) D0 a0 \<pi>0"
         using assms(1) has_as_product_def by blast
       obtain \<pi>1 where \<pi>1: "product_cone J1 (\<cdot>) D1 a1 \<pi>1"
         using assms(2) has_as_product_def by blast
       interpret J0: category J0
         using \<pi>0 product_cone.axioms(1) by metis
       interpret J1: category J1
         using \<pi>1 product_cone.axioms(1) by metis
       interpret D0: discrete_diagram J0 C D0
         using \<pi>0 product_cone.axioms(3) by metis
       interpret D1: discrete_diagram J1 C D1
         using \<pi>1 product_cone.axioms(3) by metis
       interpret \<pi>0: product_cone J0 C D0 a0 \<pi>0
         using \<pi>0 by auto
       interpret \<pi>1: product_cone J1 C D1 a1 \<pi>1
         using \<pi>1 by auto
       interpret J: discrete_category \<open>Collect J0.arr \<union> Collect J1.arr\<close> J0.null
         using J0.not_arr_null assms(5) by unfold_locales auto
       interpret X: binary_product_shape .
       interpret a0xa1: binary_product_diagram C a0 a1
         using assms(3) has_as_binary_product_def
         by (simp add: binary_product_diagram.intro binary_product_diagram_axioms.intro
             category_axioms)
       have p0p1: "a0xa1.has_as_binary_product p0 p1"
         using assms(3) has_as_binary_product_def [of a0 a1 p0 p1] by simp
 
       let ?D = "(\<lambda>i. if i \<in> Collect J0.arr then D0 i
                      else if i \<in> Collect J1.arr then D1 i
                      else null)"
       let ?a = "dom p0"
       let ?\<pi> = "\<lambda>i. if i \<in> Collect J0.arr then \<pi>0 i \<cdot> p0
                     else if i \<in> Collect J1.arr then \<pi>1 i \<cdot> p1
                     else null"
 
       let ?p0p1 = "a0xa1.mkCone p0 p1"
       interpret p0p1: limit_cone X.comp C a0xa1.map ?a ?p0p1
         using p0p1 by simp
       have a: "ide ?a"
         using p0p1.ide_apex by simp
       have p0: "\<guillemotleft>p0 : ?a \<rightarrow> a0\<guillemotright>"
         using a0xa1.mkCone_def p0p1.preserves_hom [of X.FF X.FF X.FF] X.ide_char X.ide_in_hom
         by auto
       have p1: "\<guillemotleft>p1 : ?a \<rightarrow> a1\<guillemotright>"
         using a0xa1.mkCone_def p0p1.preserves_hom [of X.TT X.TT X.TT] X.ide_char X.ide_in_hom
         by auto
 
       interpret D: discrete_diagram J.comp C ?D
         using assms J.arr_char J.dom_char J.cod_char J.is_discrete D0.is_discrete D1.is_discrete
               J.cod_comp J.seq_char
         by unfold_locales auto
       interpret A: constant_functor J.comp C ?a
         using p0p1.ide_apex by unfold_locales simp
       interpret \<pi>: natural_transformation J.comp C A.map ?D ?\<pi>
       proof
         fix j
         show "\<not> J.arr j \<Longrightarrow> ?\<pi> j = null"
           by simp
         assume j: "J.arr j"
         have \<pi>0j: "J0.arr j \<Longrightarrow> \<guillemotleft>\<pi>0 j : a0 \<rightarrow> D0 j\<guillemotright>"
           using D0.is_discrete by auto
         have \<pi>1j: "J1.arr j \<Longrightarrow> \<guillemotleft>\<pi>1 j : a1 \<rightarrow> D1 j\<guillemotright>"
           using D1.is_discrete by auto
         show "dom (?\<pi> j) = A.map (J.dom j)"
           using j J.arr_char p0 p1 \<pi>0j \<pi>1j
           by fastforce
         show "cod (?\<pi> j) = ?D (J.cod j)"
           using j J.arr_char p0 p1 \<pi>0j \<pi>1j
           by fastforce
         show "?D j \<cdot> ?\<pi> (J.dom j) = ?\<pi> j"
         proof -
           have 0: "J0.arr j \<Longrightarrow> D0 j \<cdot> \<pi>0 j \<cdot> p0 = \<pi>0 j \<cdot> p0"
           proof -
             have "J0.arr j \<Longrightarrow> (D0 j \<cdot> \<pi>0 j) \<cdot> p0 = \<pi>0 j \<cdot> p0"
               using p0 \<pi>0.is_natural_1 \<pi>0.is_natural_2 D0.is_discrete by simp
             thus "J0.arr j \<Longrightarrow> D0 j \<cdot> \<pi>0 j \<cdot> p0 = \<pi>0 j \<cdot> p0"
               using comp_assoc by simp
           qed
           have 1: "J1.arr j \<Longrightarrow> D1 j \<cdot> \<pi>1 j \<cdot> p1 = \<pi>1 j \<cdot> p1"
           proof -
             have "J1.arr j \<Longrightarrow> (D1 j \<cdot> \<pi>1 j) \<cdot> p1 = \<pi>1 j \<cdot> p1"
               using p1 \<pi>1.is_natural_1 \<pi>1.is_natural_2 D1.is_discrete by simp
             thus "J1.arr j \<Longrightarrow> D1 j \<cdot> \<pi>1 j \<cdot> p1 = \<pi>1 j \<cdot> p1"
               using comp_assoc by simp
           qed
           show ?thesis
             using 0 1 by auto
         qed
         show "?\<pi> (J.cod j) \<cdot> A.map j = ?\<pi> j"
           using j comp_arr_dom p0 p1 comp_assoc by auto
       qed
       interpret \<pi>: cone J.comp C ?D ?a ?\<pi> ..
       interpret \<pi>: product_cone J.comp C ?D ?a ?\<pi>
       proof
         show "\<And>a' \<chi>'. D.cone a' \<chi>' \<Longrightarrow> \<exists>!f. \<guillemotleft>f : a' \<rightarrow> ?a\<guillemotright> \<and> D.cones_map f ?\<pi> = \<chi>'"
         proof -
           fix a' \<chi>'
           assume \<chi>': "D.cone a' \<chi>'"
           interpret \<chi>': cone J.comp C ?D a' \<chi>'
             using \<chi>' by simp
           show "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> ?a\<guillemotright> \<and> D.cones_map f ?\<pi> = \<chi>'"
           proof
             let ?\<chi>0' = "\<lambda>i. if i \<in> Collect J0.arr then \<chi>' i else null"
             let ?\<chi>1' = "\<lambda>i. if i \<in> Collect J1.arr then \<chi>' i else null"
             have 0: "\<And>i. i \<in> Collect J0.arr \<Longrightarrow> \<chi>' i \<in> hom a' (D0 i)"
               using J.arr_char by auto
             have 1: "\<And>i. i \<in> Collect J1.arr \<Longrightarrow> \<chi>' i \<in> hom a' (D1 i)"
               using J.arr_char `Collect J0.arr \<inter> Collect J1.arr = {}` by force
             interpret A0': constant_functor J0 C a'
               apply unfold_locales using \<chi>'.ide_apex by auto
             interpret A1': constant_functor J1 C a'
               apply unfold_locales using \<chi>'.ide_apex by auto
             interpret \<chi>0': cone J0 C D0 a' ?\<chi>0'
             proof (unfold_locales)
               fix j
               show "\<not> J0.arr j \<Longrightarrow> (if j \<in> Collect J0.arr then \<chi>' j else null) = null"
                 by simp
               assume j: "J0.arr j"
               show 0: "dom (?\<chi>0' j) = A0'.map (J0.dom j)"
                 using j by simp
               show 1: "cod (?\<chi>0' j) = D0 (J0.cod j)"
                 using j J.arr_char J.cod_char D0.is_discrete by simp
               show "D0 j \<cdot> (?\<chi>0' (J0.dom j)) = ?\<chi>0' j"
                 using 1 j J.arr_char D0.is_discrete comp_cod_arr by simp
               show "?\<chi>0' (J0.cod j) \<cdot> A0'.map j = ?\<chi>0' j"
                 using 0 j J.arr_char D0.is_discrete comp_arr_dom by simp
             qed
             interpret \<chi>1': cone J1 C D1 a' ?\<chi>1'
             proof (unfold_locales)
               fix j
               show "\<not> J1.arr j \<Longrightarrow> (if j \<in> Collect J1.arr then \<chi>' j else null) = null"
                 by simp
               assume j: "J1.arr j"
               show 0: "dom (?\<chi>1' j) = A1'.map (J1.dom j)"
                 using j by simp
               show 1: "cod (?\<chi>1' j) = D1 (J1.cod j)"
                 using assms(4) j J.arr_char J.cod_char D1.is_discrete by auto
               show "D1 j \<cdot> (?\<chi>1' (J1.dom j)) = ?\<chi>1' j"
                 using 1 j J.arr_char D1.is_discrete comp_cod_arr by simp
               show "?\<chi>1' (J1.cod j) \<cdot> A1'.map j = ?\<chi>1' j"
                 using 0 j J.arr_char D1.is_discrete comp_arr_dom by simp
             qed
             define f0 where "f0 = \<pi>0.induced_arrow a' ?\<chi>0'"
             define f1 where "f1 = \<pi>1.induced_arrow a' ?\<chi>1'"
             have f0: "\<guillemotleft>f0 : a' \<rightarrow> a0\<guillemotright>"
               using f0_def \<pi>0.induced_arrowI \<chi>0'.cone_axioms by simp
             have f1: "\<guillemotleft>f1 : a' \<rightarrow> a1\<guillemotright>"
               using f1_def \<pi>1.induced_arrowI \<chi>1'.cone_axioms by simp
             have 2: "a0xa1.is_rendered_commutative_by f0 f1"
               using f0 f1 by auto
 
             interpret p0p1: binary_product_cone C a0 a1 p0 p1 ..
             interpret f0f1: cone X.comp C a0xa1.map a' \<open>a0xa1.mkCone f0 f1\<close>
               using 2 f0 f1 a0xa1.cone_mkCone [of f0 f1] by auto
             define f where "f = p0p1.induced_arrow a' (a0xa1.mkCone f0 f1)"
 
             have f: "\<guillemotleft>f : a' \<rightarrow> ?a\<guillemotright>"
               using f_def 2 f0 f1 p0p1.induced_arrowI'(1) by auto
             moreover have \<chi>': "D.cones_map f ?\<pi> = \<chi>'"
             proof
               fix j
               show "D.cones_map f ?\<pi> j = \<chi>' j"
               proof (cases "J0.arr j", cases "J1.arr j")
                 show "\<lbrakk>J0.arr j; J1.arr j\<rbrakk> \<Longrightarrow> D.cones_map f ?\<pi> j = \<chi>' j"
                   using assms(4) by auto
                 show "\<lbrakk>J0.arr j; \<not> J1.arr j\<rbrakk> \<Longrightarrow> D.cones_map f ?\<pi> j = \<chi>' j"
                 proof -
                   assume J0: "J0.arr j" and J1: "\<not> J1.arr j"
                   have "D.cones_map f ?\<pi> j = (\<pi>0 j \<cdot> p0) \<cdot> f"
                     using f J0 J1 \<pi>.cone_axioms by auto
                   also have "... = \<pi>0 j \<cdot> p0 \<cdot> f"
                     using comp_assoc by simp
                   also have "... = \<pi>0 j \<cdot> f0"
                     using 2 f0 f1 f_def p0p1.induced_arrowI' by auto
                   also have "... = \<chi>' j"
                   proof -
                     have "\<pi>0 j \<cdot> f0 = \<pi>0 j \<cdot> \<pi>0.induced_arrow' a' \<chi>'"
                       unfolding f0_def by simp
                     also have "... = (\<lambda>j. if J0.arr j then
                                             \<pi>0 j \<cdot> \<pi>0.induced_arrow a'
                                                     (\<lambda>i. if i \<in> Collect J0.arr then \<chi>' i else null)
                                           else null) j"
                       using J0 by simp
                     also have "... = D0.mkCone \<chi>' j"
                     proof -
                       have "(\<lambda>j. if J0.arr j then
                                     \<pi>0 j \<cdot> \<pi>0.induced_arrow a'
                                              (\<lambda>i. if i \<in> Collect J0.arr then \<chi>' i else null)
                                  else null) =
                             D0.mkCone \<chi>'"
                         using f0 f0_def \<pi>0.induced_arrowI(2) [of ?\<chi>0' a'] J0
                               D0.mkCone_cone \<chi>0'.cone_axioms \<pi>0.cone_axioms J0
                         by auto
                       thus ?thesis by meson
                     qed
                     also have "... = \<chi>' j"
                       using J0 by simp
                     finally show ?thesis by blast
                   qed
                   finally show ?thesis by simp
                 qed
                 show "\<not> J0.arr j \<Longrightarrow> D.cones_map f ?\<pi> j = \<chi>' j"
                 proof (cases "J1.arr j")
                   show "\<lbrakk>\<not> J0.arr j; \<not> J1.arr j\<rbrakk> \<Longrightarrow> D.cones_map f ?\<pi> j = \<chi>' j"
                     using f \<pi>.cone_axioms \<chi>'.is_extensional by auto
                   show "\<lbrakk>\<not> J0.arr j; J1.arr j\<rbrakk> \<Longrightarrow> D.cones_map f ?\<pi> j = \<chi>' j"
                   proof -
                     assume J0: "\<not> J0.arr j" and J1: "J1.arr j"
                     have "D.cones_map f ?\<pi> j = (\<pi>1 j \<cdot> p1) \<cdot> f"
                       using J0 J1 f \<pi>.cone_axioms by auto
                     also have "... = \<pi>1 j \<cdot> p1 \<cdot> f"
                       using comp_assoc by simp
                     also have "... = \<pi>1 j \<cdot> f1"
                       using 2 f0 f1 f_def p0p1.induced_arrowI' by auto
                     also have "... = \<chi>' j"
                     proof -
                       have "\<pi>1 j \<cdot> f1 = \<pi>1 j \<cdot> \<pi>1.induced_arrow' a' \<chi>'"
                         unfolding f1_def by simp
                       also have "... = (\<lambda>j. if J1.arr j then
                                               \<pi>1 j \<cdot> \<pi>1.induced_arrow a'
                                                       (\<lambda>i. if i \<in> Collect J1.arr
                                                            then \<chi>' i else null)
                                             else null) j"
                         using J1 by simp
                       also have "... = D1.mkCone \<chi>' j"
                       proof -
                         have "(\<lambda>j. if J1.arr j then
                                       \<pi>1 j \<cdot> \<pi>1.induced_arrow a'
                                                (\<lambda>i. if i \<in> Collect J1.arr then \<chi>' i else null)
                                    else null) =
                               D1.mkCone \<chi>'"
                           using f1 f1_def \<pi>1.induced_arrowI(2) [of ?\<chi>1' a'] J1
                                 D1.mkCone_cone [of a' \<chi>'] \<chi>1'.cone_axioms \<pi>1.cone_axioms J1
                           by auto
                         thus ?thesis by meson
                       qed
                       also have "... = \<chi>' j"
                         using J1 by simp
                       finally show ?thesis by blast
                     qed
                     finally show ?thesis by simp
                   qed
                 qed
               qed
             qed
             ultimately show "\<guillemotleft>f : a' \<rightarrow> ?a\<guillemotright> \<and> D.cones_map f ?\<pi> = \<chi>'" by blast
             show "\<And>f'. \<guillemotleft>f' : a' \<rightarrow> ?a\<guillemotright> \<and> D.cones_map f' ?\<pi> = \<chi>' \<Longrightarrow> f' = f"
             proof -
               fix f'
               assume f': "\<guillemotleft>f' : a' \<rightarrow> ?a\<guillemotright> \<and> D.cones_map f' ?\<pi> = \<chi>'"
               let ?f0' = "p0 \<cdot> f'"
               let ?f1' = "p1 \<cdot> f'"
               have 1: "a0xa1.is_rendered_commutative_by ?f0' ?f1'"
                 using f' p0 p1 p0p1.renders_commutative seqI' by auto
               have f0': "\<guillemotleft>?f0' : a' \<rightarrow> a0\<guillemotright>"
                 using f' p0 by auto
               have f1': "\<guillemotleft>?f1' : a' \<rightarrow> a1\<guillemotright>"
                 using f' p1 by auto
               have "p0 \<cdot> f = p0 \<cdot> f'"
               proof -
                 have "D0.cones_map (p0 \<cdot> f) \<pi>0 = ?\<chi>0'"
                   using f p0 \<pi>0.cone_axioms \<chi>' \<pi>.cone_axioms comp_assoc assms(4) seqI'
                   by fastforce
                 moreover have "D0.cones_map (p0 \<cdot> f') \<pi>0 = ?\<chi>0'"
                   using f' p0 \<pi>0.cone_axioms \<pi>.cone_axioms comp_assoc assms(4) seqI'
                   by fastforce
                 moreover have "p0 \<cdot> f = f0"
                   using 2 f0 f_def p0p1.induced_arrowI'(2) by blast
                 ultimately show ?thesis
                   using f0 f0' \<chi>0'.cone_axioms \<pi>0.is_universal [of a'] by auto
               qed
               moreover have "p1 \<cdot> f = p1 \<cdot> f'"
               proof -
                 have "D1.cones_map (p1 \<cdot> f) \<pi>1 = ?\<chi>1'"
                 proof
                   fix j
                   show "D1.cones_map (p1 \<cdot> f) \<pi>1 j = ?\<chi>1' j"
                     using f p1 \<pi>1.cone_axioms \<chi>' \<pi>.cone_axioms comp_assoc assms(4) seqI'
                     apply auto
                     by auto
                 qed
                 moreover have "D1.cones_map (p1 \<cdot> f') \<pi>1 = ?\<chi>1'"
                 proof
                   fix j
                   show "D1.cones_map (p1 \<cdot> f') \<pi>1 j = ?\<chi>1' j"
                     using f' p1 \<pi>1.cone_axioms \<pi>.cone_axioms comp_assoc assms(4) seqI'
                     apply auto
                     by auto
                 qed
                 moreover have "p1 \<cdot> f = f1"
                   using 2 f1 f_def p0p1.induced_arrowI'(3) by blast
                   ultimately show ?thesis
                 using f1 f1' \<chi>1'.cone_axioms \<pi>1.is_universal [of a'] by auto
               qed
               ultimately show "f' = f"
                 using f f' p0p1.is_universal' [of a']
                 by (metis (no_types, lifting) "1" dom_comp in_homE p0p1.is_universal' p1 seqI')
             qed
           qed
         qed
       qed
       show "has_as_product J.comp ?D ?a"
         unfolding has_as_product_def
         using \<pi>.product_cone_axioms by auto
     qed
 
   end
 
   sublocale cartesian_category \<subseteq> category_with_finite_products
   proof
     obtain t where t: "terminal t" using has_terminal by blast
     { fix n :: nat
       have "\<And>I :: nat set. finite I \<and> card I = n \<Longrightarrow> has_products I"
       proof (induct n)
         show "\<And>I :: nat set. finite I \<and> card I = 0 \<Longrightarrow> has_products I"
         proof -
           fix I :: "nat set"
           assume "finite I \<and> card I = 0"
           hence I: "I = {}" by force
           thus "has_products I"
           proof -
             interpret J: discrete_category I 0
               apply unfold_locales using I by auto
             have "\<And>D. discrete_diagram J.comp C D \<Longrightarrow> \<exists>a. has_as_product J.comp D a"
             proof -
               fix D
               assume D: "discrete_diagram J.comp C D"
               interpret D: discrete_diagram J.comp C D using D by auto
               interpret D: empty_diagram J.comp C D
                  using I J.arr_char by unfold_locales simp
               have "has_as_product J.comp D t"
                 using t D.has_as_limit_iff_terminal has_as_product_def product_cone_def
                       J.category_axioms category_axioms D.discrete_diagram_axioms
                 by metis
               thus "\<exists>a. has_as_product J.comp D a" by blast
             qed
             moreover have "I \<noteq> UNIV"
               using I by blast
             ultimately show ?thesis
               using I has_products_def
               by (metis category_with_terminal_object.has_terminal discrete_diagram.product_coneI
                   discrete_diagram_def empty_diagram.has_as_limit_iff_terminal empty_diagram.intro
                   empty_diagram_axioms.intro empty_iff has_as_product_def
                   is_category_with_terminal_object mem_Collect_eq)
           qed
         qed
         show "\<And>n I :: nat set.
                 \<lbrakk> (\<And>I :: nat set. finite I \<and> card I = n \<Longrightarrow> has_products I);
                   finite I \<and> card I = Suc n \<rbrakk>
                     \<Longrightarrow> has_products I"
         proof -
           fix n :: nat
           fix I :: "nat set"
           assume IH: "\<And>I :: nat set. finite I \<and> card I = n \<Longrightarrow> has_products I"
           assume I: "finite I \<and> card I = Suc n"
           show "has_products I"
           proof -
             have "card I = 1 \<Longrightarrow> has_products I"
               using I has_unary_products by blast
             moreover have "card I \<noteq> 1 \<Longrightarrow> has_products I"
             proof -
               assume "card I \<noteq> 1"
               hence cardI: "card I > 1" using I by simp
               obtain i where i: "i \<in> I" using cardI by fastforce
               let ?I0 = "{i}" and ?I1 = "I - {i}"
               have 1: "I = ?I0 \<union> ?I1 \<and> ?I0 \<inter> ?I1 = {} \<and> card ?I0 = 1 \<and> card ?I1 = n"
                 using i I cardI by auto
               show "has_products I"
               proof (unfold has_products_def, intro conjI allI impI)
                 show "I \<noteq> UNIV"
                   using I by auto
                 fix J D
                 assume D: "discrete_diagram J C D \<and> Collect (partial_magma.arr J) = I"
                 interpret D: discrete_diagram J C D
                   using D by simp
                 have Null: "D.J.null \<notin> ?I0 \<and> D.J.null \<notin> ?I1"
                   using D D.J.not_arr_null i by blast
                 interpret J0: discrete_category ?I0 D.J.null
                   using 1 Null D by unfold_locales auto
                 interpret J1: discrete_category ?I1 D.J.null
                   using Null by unfold_locales auto
                 interpret J0uJ1: discrete_category \<open>Collect J0.arr \<union> Collect J1.arr\<close> J0.null
                   using Null 1 J0.null_char J1.null_char by unfold_locales auto
                 interpret D0: discrete_diagram_from_map ?I0 C D D.J.null
                   using 1 J0.ide_char D.preserves_ide D D.is_discrete i by unfold_locales auto
                 interpret D1: discrete_diagram_from_map ?I1 C D D.J.null
                   using 1 J1.ide_char D.preserves_ide D D.is_discrete i by unfold_locales auto
                 obtain a0 where a0: "has_as_product J0.comp D0.map a0"
                   using 1 has_unary_products [of ?I0] has_products_def [of ?I0]
                         D0.discrete_diagram_axioms
                   by fastforce
                 obtain a1 where a1: "has_as_product J1.comp D1.map a1"
                   using 1 I IH [of ?I1] has_products_def [of ?I1] D1.discrete_diagram_axioms
                   by blast
                 have 2: "\<exists>p0 p1. has_as_binary_product a0 a1 p0 p1"
                 proof -
                   have "ide a0 \<and> ide a1"
                     using a0 a1 product_is_ide by auto
                   thus ?thesis
                      using a0 a1 has_binary_products has_binary_products_def by simp
                 qed
                 obtain p0 p1 where a: "has_as_binary_product a0 a1 p0 p1"
                   using 2 by auto
                 let ?a = "dom p0"
                 have "has_as_product J D ?a"
                 proof -
                   have "D = (\<lambda>j. if j \<in> Collect J0.arr then D0.map j
                                  else if j \<in> Collect J1.arr then D1.map j
                                  else null)"
                   proof
                     fix j
                     show "D j = (if j \<in> Collect J0.arr then D0.map j
                                  else if j \<in> Collect J1.arr then D1.map j
                                  else null)"
                       using 1 D0.map_def D1.map_def D.is_extensional D J0.arr_char J1.arr_char
                       by auto
                   qed
                   moreover have "J = J0uJ1.comp"
                   proof -
                     have "\<And>j j'. J j j' = J0uJ1.comp j j'"
                     proof -
                       fix j j'
                       show "J j j' = J0uJ1.comp j j'"
                         using D J0uJ1.arr_char J0.arr_char J1.arr_char D.is_discrete i
                         apply (cases "j \<in> ?I0", cases "j' \<in> ?I0")
                           apply simp_all
                           apply auto[1]
                          apply (metis D.J.comp_arr_ide D.J.comp_ide_arr D.J.ext D.J.seqE
                             D.is_discrete J0.null_char J0uJ1.null_char)
                         by (metis D.J.comp_arr_ide D.J.comp_ide_arr D.J.comp_ide_self
                             D.J.ext D.J.seqE D.is_discrete J0.null_char J0uJ1.null_char
                             mem_Collect_eq)
                     qed
                     thus ?thesis by blast
                   qed
                   moreover have "Collect J0.arr \<inter> Collect J1.arr = {}"
                     by auto
                   moreover have "J0.null = J1.null"
                     using J0.null_char J1.null_char by simp
                   ultimately show "has_as_product J D ?a"
                     using binary_product_of_products_is_product
                             [of J0.comp D0.map a0 J1.comp D1.map a1 p0 p1]
                           J0.arr_char J1.arr_char
                           1 a0 a1 a
                     by simp
                 qed
                 thus "\<exists>a. has_as_product J D a" by blast
               qed
             qed
             ultimately show "has_products I" by blast
           qed
         qed
       qed
     }
     hence 1: "\<And>n I :: nat set. finite I \<and> card I = n \<Longrightarrow> has_products I" by simp
     thus "\<And>I :: nat set. finite I \<Longrightarrow> has_products I" by blast
   qed
 
   proposition (in cartesian_category) is_category_with_finite_products:
   shows "category_with_finite_products C"
     ..
 
 end
 
diff --git a/thys/Category3/CategoryWithFiniteLimits.thy b/thys/Category3/CategoryWithFiniteLimits.thy
--- a/thys/Category3/CategoryWithFiniteLimits.thy
+++ b/thys/Category3/CategoryWithFiniteLimits.thy
@@ -1,424 +1,424 @@
 (*  Title:       CategoryWithFiniteLimits
     Author:      Eugene W. Stark <stark@cs.stonybrook.edu>, 2020
     Maintainer:  Eugene W. Stark <stark@cs.stonybrook.edu>
 *)
 
 chapter "Category with Finite Limits"
 
 theory CategoryWithFiniteLimits
-imports Category3.CartesianCategory Category3.CategoryWithPullbacks
+imports CartesianCategory CategoryWithPullbacks
 begin
 
   text\<open>
     In this chapter we define ``category with finite limits'' and show that such
     categories coincide with those having pullbacks and a terminal object.
 
     Since we can't quantify over types in HOL, the best we can do at defining the notion
     ``category with finite limits'' is to state it for a fixed choice of type (e.g.~@{typ nat})
     for the arrows of the ``diagram shape''.  However, we then have to go to some
     trouble to show the existence of finite limits for diagram shapes at other types.
   \<close>
 
   locale category_with_finite_limits =
     category +
   assumes has_finite_limits:
             "\<lbrakk> category (J :: nat comp); finite (Collect (partial_magma.arr J)) \<rbrakk>
                  \<Longrightarrow> has_limits_of_shape J"
   begin
 
     text\<open>
       We show that a category with finite limits has pullbacks and a terminal object
       and is therefore also a cartesian category.
     \<close>
 
     interpretation category_with_pullbacks C
     proof -
       interpret J: cospan_shape
         by unfold_locales
       have 1: "finite (Collect J.arr)"
       proof -
         have "Collect J.arr = {J.AA, J.BB, J.TT, J.AT, J.BT}"
           using J.arr_char cospan_shape.Dom.cases by auto
         thus ?thesis by simp
       qed
       obtain J' :: "nat comp" where J': "isomorphic_categories J' J.comp"
         using 1 J.finite_imp_ex_iso_nat_comp by blast
       interpret J'J: isomorphic_categories J' J.comp
         using J' by simp
       obtain \<phi> \<psi> where \<phi>\<psi>: "inverse_functors J.comp J' \<phi> \<psi>"
         using J'J.iso inverse_functors_sym by blast
       interpret \<phi>\<psi>: inverse_functors J.comp J' \<phi> \<psi>
         using \<phi>\<psi> by simp
       interpret \<psi>: invertible_functor J.comp J' \<psi>
         using \<phi>\<psi>.inverse_functors_axioms
         by unfold_locales auto
       show "category_with_pullbacks C"
       proof
         show "has_pullbacks"
         proof (unfold has_pullbacks_def has_as_pullback_def, intro allI impI)
           fix f0 f1
           assume cospan: "cospan f0 f1"
           interpret D: cospan_diagram C f0 f1
             using cospan
             by (simp add: category_axioms cospan_diagram_axioms_def cospan_diagram_def)
           have 2: "has_limits_of_shape J.comp"
             using 1 bij_betw_finite J'J.A.category_axioms has_finite_limits \<psi>.bij_betw_arr_sets
                   has_limits_preserved_by_isomorphism J'J.isomorphic_categories_axioms
             by blast
           obtain a \<chi> where \<chi>: "limit_cone J.comp C D.map a \<chi>"
             using 2 D.diagram_axioms has_limits_of_shape_def by blast
           interpret \<chi>: limit_cone J.comp C D.map a \<chi>
             using \<chi> by simp
           have "D.map = cospan_diagram.map C f0 f1" by simp
           moreover have "a = dom (\<chi> J.AA)"
             using J.arr_char \<chi>.component_in_hom by force
           moreover have "\<chi> = cospan_diagram.mkCone (\<cdot>) f0 f1 (\<chi> J.AA) (\<chi> J.BB)"
             using D.mkCone_cone \<chi>.cone_axioms by auto
           ultimately have "limit_cone (\<cdot>\<^sub>J) (\<cdot>)
                              (cospan_diagram.map (\<cdot>) f0 f1) (dom (\<chi> J.AA))
                              (cospan_diagram.mkCone (\<cdot>) f0 f1 (\<chi> J.AA) (\<chi> J.BB))"
             using \<chi>.limit_cone_axioms by simp
           thus "\<exists>p0 p1. cospan f0 f1 \<and>
                         limit_cone (\<cdot>\<^sub>J) (\<cdot>)
                           (cospan_diagram.map (\<cdot>) f0 f1) (dom p0)
                           (cospan_diagram.mkCone (\<cdot>) f0 f1 p0 p1)"
             using cospan by auto
         qed
       qed
     qed
 
     lemma is_category_with_pullbacks:
     shows "category_with_pullbacks C"
       ..
 
     sublocale category_with_pullbacks C ..
 
     interpretation category_with_terminal_object C
     proof
       show "\<exists>a. terminal a"
       proof -
         interpret J: discrete_category \<open>{} :: nat set\<close> 0
           by unfold_locales simp
         have 1: "has_limits_of_shape J.comp"
           using has_finite_limits
           by (metis Collect_empty_eq J.arr_char J.is_category empty_iff finite.emptyI)
         interpret D: diagram J.comp C \<open>\<lambda>_. null\<close>
           by unfold_locales auto
         obtain t \<tau> where \<tau>: "D.limit_cone t \<tau>"
           using 1 D.diagram_axioms has_limits_of_shape_def by blast
         interpret \<tau>: limit_cone J.comp C \<open>\<lambda>_. null\<close> t \<tau>
           using \<tau> by simp
         have "terminal t"
         proof
           show "ide t"
             using \<tau>.ide_apex by simp
           fix a
           assume a: "ide a"
           show "\<exists>!f. \<guillemotleft>f : a \<rightarrow> t\<guillemotright>"
           proof -
             interpret a: constant_functor J.comp C a
               using a by unfold_locales
             interpret \<chi>: cone J.comp C \<open>\<lambda>_.null\<close> a \<open>\<lambda>_.null\<close>
               apply unfold_locales
                   apply simp
               using dom_null cod_null comp_null
               by blast+
             have "\<exists>!f. \<guillemotleft>f : a \<rightarrow> t\<guillemotright> \<and> D.cones_map f \<tau> = (\<lambda>_. null)"
               using \<tau>.induced_arrowI [of "\<lambda>_.null" a] \<chi>.cone_axioms
                     \<tau>.is_universal [of a "\<lambda>_. null"]
               by simp
             moreover have "\<And>f. \<guillemotleft>f : a \<rightarrow> t\<guillemotright> \<Longrightarrow> D.cones_map f \<tau> = (\<lambda>_. null)"
               using \<tau>.cone_axioms by auto
             ultimately show ?thesis by auto
           qed
         qed
         thus ?thesis by blast
       qed
     qed
 
     lemma is_category_with_terminal_object:
     shows "category_with_terminal_object C"
       ..
 
     sublocale category_with_terminal_object C ..
 
     sublocale category_with_finite_products
       using has_finite_limits has_finite_products_if_has_finite_limits
             has_limits_of_shape_def diagram_def
       by unfold_locales blast
 
     sublocale cartesian_category ..
 
   end
 
   locale category_with_pullbacks_and_terminal =
     category_with_pullbacks +
     category_with_terminal_object
 
   sublocale category_with_finite_limits \<subseteq> category_with_pullbacks_and_terminal ..
 
   text\<open>
     Conversely, we show that a category with pullbacks and a terminal object also
     has finite products and equalizers, and therefore has finite limits.
   \<close>
 
   context category_with_pullbacks_and_terminal
   begin
 
     interpretation ECP: elementary_category_with_pullbacks C prj0 prj1
       using extends_to_elementary_category_with_pullbacks by simp
 
     abbreviation prj0'
     where "prj0' a b \<equiv> (if ide a \<and> ide b then prj0 (trm a) (trm b) else null)"
 
     abbreviation prj1'
     where "prj1' a b \<equiv> (if ide a \<and> ide b then prj1 (trm a) (trm b) else null)"
 
     interpretation ECC: elementary_cartesian_category C prj0' prj1' \<one> trm
       using trm_naturality ECP.universal
       by unfold_locales auto
 
     interpretation category_with_equalizers C
     proof (unfold_locales, unfold has_equalizers_def, intro allI impI)
       fix f0 f1
       assume par: "par f0 f1"
       interpret J: parallel_pair
         by unfold_locales
       interpret D: parallel_pair_diagram C f0 f1
         using par by unfold_locales auto
       have 1: "cospan (ECC.prod f1 (dom f0)) (ECC.prod f0 (dom f0))"
         using par by simp
       let ?g0 = "ECC.prod f0 (dom f0) \<cdot> ECC.dup (dom f0)"
       let ?g1 = "ECC.prod f1 (dom f1) \<cdot> ECC.dup (dom f1)"
       have g0: "\<guillemotleft>?g0 : dom f0 \<rightarrow> ECC.prod (cod f0) (dom f0)\<guillemotright>"
         using par by simp
       have g1: "\<guillemotleft>?g1 : dom f1 \<rightarrow> ECC.prod (cod f1) (dom f1)\<guillemotright>"
         using par by simp
       define e0 where "e0 = prj0 ?g1 ?g0"
       define e1 where "e1 = prj1 ?g1 ?g0"
       have e0: "\<guillemotleft>e0 : dom e0 \<rightarrow> dom f0\<guillemotright>"
         using par 1 e0_def by auto
       have e1: "\<guillemotleft>e1 : dom e0 \<rightarrow> dom f1\<guillemotright>"
         using par 1 e1_def e0_def by auto
       have eq: "e0 = e1"
       proof -
         have "e1 = prj0' (cod f1) (dom f1) \<cdot> ?g1 \<cdot> e1"
         proof -
           have "((prj0' (cod f1) (dom f1) \<cdot> (ECC.prod f1 (dom f1))) \<cdot> ECC.dup (dom f1)) \<cdot> e1 =
                 dom f1 \<cdot> e1"
             using par ECC.pr_naturality(1) [of "dom f1" "dom f1" "dom f1" f1 "dom f1" "cod f1"]
                   comp_cod_arr ECC.pr_dup(1)
             by auto
           also have "... = e1"
             using par e1 comp_cod_arr by blast
           finally show ?thesis
             using comp_assoc by simp
         qed
         also have "... = prj0' (cod f1) (dom f1) \<cdot> ?g0 \<cdot> e0"
           using par ECP.pullback_commutes
           unfolding commutative_square_def e0_def e1_def by simp
         also have "... = e0"
         proof -
           have "((prj0' (cod f1) (dom f1) \<cdot> (ECC.prod f0 (dom f0))) \<cdot> ECC.dup (dom f0)) \<cdot> e0 =
                 dom f0 \<cdot> e0"
             using par ECC.pr_naturality(1) [of "dom f0" "dom f0" "dom f1" f0 "dom f0" "cod f0"]
                   comp_cod_arr ECC.pr_dup(1) ide_dom
             by auto
           also have "... = e0"
             using e0 comp_cod_arr by blast
           finally show ?thesis
             using comp_assoc by simp
         qed
         finally show ?thesis by auto
       qed
       have equalizes: "D.is_equalized_by e0"
       proof
         show "seq f0 e0"
           using par e0 by auto
         show "f0 \<cdot> e0 = f1 \<cdot> e0"
         proof -
           have "f0 \<cdot> e0 = (f0 \<cdot> dom f0) \<cdot> e0"
             using par comp_arr_dom by simp
           also have "... = (f0 \<cdot> (prj1' (dom f0) (dom f0) \<cdot> ECC.dup (dom f0))) \<cdot> e0"
             using par ECC.pr_dup(2) by auto
           also have "... = ((f0 \<cdot> prj1' (dom f0) (dom f0)) \<cdot> ECC.dup (dom f0)) \<cdot> e0"
             using comp_assoc by auto
           also have "... = prj1' (cod f1) (dom f1) \<cdot> ?g0 \<cdot> e0"
             using par ECC.pr_naturality(2) [of "dom f0" "dom f0" "dom f1" f0 "dom f0" "cod f0"]
             by (metis (no_types, lifting) arr_dom cod_dom dom_dom comp_assoc)
           also have "... = prj1' (cod f1) (dom f1) \<cdot> ?g1 \<cdot> e1"
             using par ECP.pullback_commutes [of ?g1 ?g0]
             unfolding commutative_square_def e0_def e1_def by simp
           also have "... = (prj1' (cod f1) (dom f1) \<cdot> ?g1) \<cdot> e1"
             using comp_assoc by simp
           also have "... = (f1 \<cdot> (prj1' (dom f1) (dom f1) \<cdot> ECC.dup (dom f1))) \<cdot> e1"
             using par ECC.pr_naturality(2) [of "dom f1" "dom f1" "dom f1" f1 "dom f1" "cod f1"]
             by (metis (no_types, lifting) arr_dom cod_dom dom_dom comp_assoc)
           also have "... = (f1 \<cdot> dom f1) \<cdot> e1"
             using par ECC.pr_dup(2) by auto
           also have "... = f1 \<cdot> e1"
             using par comp_arr_dom by simp
           also have "... = f1 \<cdot> e0"
             using eq by simp
           finally show ?thesis by simp
         qed
       qed
       show "\<exists>e. has_as_equalizer f0 f1 e"
       proof
         interpret E: constant_functor J.comp C \<open>dom e0\<close>
           using par e0 by unfold_locales auto
         interpret \<chi>: cone J.comp C D.map \<open>dom e0\<close> \<open>D.mkCone e0\<close>
           using equalizes D.cone_mkCone e0_def by auto
         interpret \<chi>: limit_cone J.comp C D.map \<open>dom e0\<close> \<open>D.mkCone e0\<close>
         proof
           show "\<And>a' \<chi>'. D.cone a' \<chi>' \<Longrightarrow>
                          \<exists>!f. \<guillemotleft>f : a' \<rightarrow> dom e0\<guillemotright> \<and> D.cones_map f (D.mkCone e0) = \<chi>'"
           proof -
             fix a' \<chi>'
             assume \<chi>': "D.cone a' \<chi>'"
             interpret \<chi>': cone J.comp C D.map a' \<chi>'
               using \<chi>' by simp
             have 3: "commutative_square ?g1 ?g0 (\<chi>' J.Zero) (\<chi>' J.Zero)"
             proof
               show "cospan ?g1 ?g0"
                 using par g0 g1 by simp
               show 4: "span (\<chi>' J.Zero) (\<chi>' J.Zero)"
                 using J.arr_char by simp
               show 5: "dom ?g1 = cod (\<chi>' J.Zero)"
                 using par g1 J.arr_char D.map_def by simp
               show "?g1 \<cdot> \<chi>' J.Zero = ?g0 \<cdot> \<chi>' J.Zero"
               proof -
                 have "?g1 \<cdot> \<chi>' J.Zero = ECC.prod f1 (dom f1) \<cdot> ECC.dup (dom f1) \<cdot> \<chi>' J.Zero"
                   using comp_assoc by simp
                 also have "... = ECC.prod f1 (dom f1) \<cdot> ECC.tuple (\<chi>' J.Zero) (\<chi>' J.Zero)"
                   using par D.map_def J.arr_char comp_cod_arr by auto
                 also have "... = ECC.tuple (f1 \<cdot> \<chi>' J.Zero) (\<chi>' J.Zero)"
                   using par ECC.prod_tuple [of "\<chi>' J.Zero" "\<chi>' J.Zero" f1 "dom f1"]
                         comp_cod_arr
                   by (metis (no_types, lifting) 4 5 g1 in_homE seqI)
                 also have "... = ECC.tuple (f0 \<cdot> \<chi>' J.Zero) (\<chi>' J.Zero)"
                   using par D.is_equalized_by_cone \<chi>'.cone_axioms by auto
                 also have "... = ECC.prod f0 (dom f0) \<cdot> ECC.tuple (\<chi>' J.Zero) (\<chi>' J.Zero)"
                   using par ECC.prod_tuple [of "\<chi>' J.Zero" "\<chi>' J.Zero" f0 "dom f0"]
                         comp_cod_arr
                   by (metis (no_types, lifting) 4 5 g1 in_homE seqI)
                 also have "... = ECC.prod f0 (dom f0) \<cdot> ECC.dup (dom f0) \<cdot> \<chi>' J.Zero"
                   using par D.map_def J.arr_char comp_cod_arr by auto
                 also have "... = ?g0 \<cdot> \<chi>' J.Zero"
                   using comp_assoc by simp
                 finally show ?thesis by blast
               qed
             qed
             show "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> dom e0\<guillemotright> \<and> D.cones_map f (D.mkCone e0) = \<chi>'"
             proof
               define f where "f = ECP.tuple (\<chi>' J.Zero) ?g1 ?g0 (\<chi>' J.Zero)"
               have 4: "e0 \<cdot> f = \<chi>' J.Zero"
                 using ECP.universal by (simp add: "3" e1_def eq f_def)
               have f: "\<guillemotleft>f : a' \<rightarrow> dom e0\<guillemotright>"
               proof -
                 have "a' = dom (\<chi>' J.Zero)"
                   by (simp add: J.arr_char)
                 thus ?thesis
                   using 3 f_def e0_def g0 g1 ECP.tuple_in_hom ECP.pbdom_def by simp
               qed
               moreover have 5: "D.cones_map f (D.mkCone e0) = \<chi>'"
               proof -
                 have "\<And>j. J.arr j \<Longrightarrow> D.mkCone e0 j \<cdot> f = \<chi>' j"
                 proof -
                   fix j
                   assume j: "J.arr j"
                   show "D.mkCone e0 j \<cdot> f = \<chi>' j"
                   proof (cases "j = J.Zero")
                     case True
                     moreover have "e0 \<cdot> f = \<chi>' J.Zero"
                       using 4 by simp
                     ultimately show ?thesis
                       unfolding f_def D.mkCone_def comp_assoc
                       using J.arr_char by simp
                     next
                     case F: False
                     hence 1: "(f0 \<cdot> e0) \<cdot> f = f0 \<cdot> \<chi>' J.Zero"
                       using 4 comp_assoc by simp
                     also have "... = \<chi>' j"
                       by (metis (no_types, lifting) F D.mkCone_cone D.mkCone_def
                           \<chi>'.cone_axioms j)
                     finally show ?thesis
                       by (simp add: F D.mkCone_def j)
                   qed
                 qed
                 thus ?thesis
                   using f e0 \<chi>.cone_axioms \<chi>'.is_extensional by auto
               qed
               ultimately show "\<guillemotleft>f : a' \<rightarrow> dom e0\<guillemotright> \<and> D.cones_map f (D.mkCone e0) = \<chi>'"
                 by simp
               fix f'
               assume f': "\<guillemotleft>f' : a' \<rightarrow> dom e0\<guillemotright> \<and> D.cones_map f' (D.mkCone e0) = \<chi>'"
               show "f' = f"
               proof -
                 have "e0 \<cdot> f' = \<chi>' J.Zero"
                   using f' D.mkCone_cone D.mkCone_def \<chi>'.cone_axioms
                         comp_assoc J.arr_char \<chi>.cone_axioms
                   by auto
                 thus ?thesis
                   using f' 3 4 eq ECP.universal [of ?g1 ?g0 "e1 \<cdot> f'" "e0 \<cdot> f'"] e0_def e1_def
                   by (metis (no_types, lifting))
               qed
             qed
           qed
         qed
         show "has_as_equalizer f0 f1 e0"
         proof -
           have "par f0 f1"
             by fact
           moreover have "D.has_as_equalizer e0"
             ..
           ultimately show ?thesis
             using has_as_equalizer_def by blast
         qed
       qed
     qed
 
     interpretation category_with_finite_products C
       by (simp add: ECC.is_cartesian_category cartesian_category.is_category_with_finite_products)
 
     lemma has_finite_products:
     shows "category_with_finite_products C"
       ..
 
     lemma has_finite_limits:
     shows "category_with_finite_limits C"
     proof
       fix J :: "nat comp"
       assume J: "category J"
       interpret J: category J
         using J by simp
       assume finite: "finite (Collect J.arr)"
       show "has_limits_of_shape J"
       proof -
         have "Collect (partial_magma.ide J) \<subseteq> Collect J.arr"
           by auto
         hence 1: "finite (Collect J.ide)"
           using finite finite_subset by blast
         have "has_products (Collect (partial_magma.ide J))"
           using 1 J.ideD(1) J.not_arr_null category_with_finite_products.has_finite_products
                 is_category_with_finite_products
           by simp
         moreover have "Collect (partial_magma.ide J) \<noteq> UNIV"
           using J.not_arr_null by blast
         moreover have "Collect (partial_magma.arr J) \<noteq> UNIV"
           using J.not_arr_null by blast
         ultimately show ?thesis
           using finite 1 J.category_axioms has_limits_if_has_products
                 has_finite_products' [of "Collect J.ide"]
                 has_finite_products' [of "Collect J.arr"]
           by simp
       qed
     qed
 
     sublocale category_with_finite_limits C
       using has_finite_limits by simp
 
   end
 
 end
 
diff --git a/thys/Category3/CategoryWithPullbacks.thy b/thys/Category3/CategoryWithPullbacks.thy
--- a/thys/Category3/CategoryWithPullbacks.thy
+++ b/thys/Category3/CategoryWithPullbacks.thy
@@ -1,1114 +1,1114 @@
 (*  Title:       CategoryWithPullbacks
     Author:      Eugene W. Stark <stark@cs.stonybrook.edu>, 2019
     Maintainer:  Eugene W. Stark <stark@cs.stonybrook.edu>
 *)
 
 chapter "Category with Pullbacks"
 
 theory CategoryWithPullbacks
-imports Category3.Limit
+imports Limit
 begin
 
 text \<open>
   \sloppypar
   In this chapter, we give a traditional definition of pullbacks in a category as
   limits of cospan diagrams and we define a locale \<open>category_with_pullbacks\<close> that
   is satisfied by categories in which every cospan diagram has a limit.
   These definitions build on the general definition of limit that we gave in
   @{theory Category3.Limit}.  We then define a locale \<open>elementary_category_with_pullbacks\<close>
   that axiomatizes categories equipped with chosen functions that assign to each cospan
   a corresponding span of ``projections'', which enjoy the familiar universal property
   of a pullback.  After developing consequences of the axioms, we prove that the two
   locales are in agreement, in the sense that every interpretation of
   \<open>category_with_pullbacks\<close> extends to an interpretation of
   \<open>elementary_category_with_pullbacks\<close>, and conversely, the underlying category of
   an interpretation of \<open>elementary_category_with_pullbacks\<close> always yields an interpretation
   of \<open>category_with_pullbacks\<close>.
 \<close>
 
   section "Commutative Squares"
 
   context category
   begin
 
     text \<open>
       The following provides some useful technology for working with commutative squares.
     \<close>
 
     definition commutative_square
     where "commutative_square f g h k \<equiv> cospan f g \<and> span h k \<and> dom f = cod h \<and> f \<cdot> h = g \<cdot> k"
 
     lemma commutative_squareI [intro, simp]:
     assumes "cospan f g" and "span h k" and "dom f = cod h" and "f \<cdot> h = g \<cdot> k"
     shows "commutative_square f g h k"
       using assms commutative_square_def by auto
 
     lemma commutative_squareE [elim]:
     assumes "commutative_square f g h k"
     and "\<lbrakk> arr f; arr g; arr h; arr k; cod f = cod g; dom h = dom k; dom f = cod h;
            dom g = cod k; f \<cdot> h = g \<cdot> k \<rbrakk> \<Longrightarrow> T"
     shows T
       using assms commutative_square_def
       by (metis (mono_tags, lifting) seqE seqI)
 
     lemma commutative_square_comp_arr:
     assumes "commutative_square f g h k" and "seq h l"
     shows "commutative_square f g (h \<cdot> l) (k \<cdot> l)"
       using assms
       apply (elim commutative_squareE, intro commutative_squareI, auto)
       using comp_assoc by metis
 
     lemma arr_comp_commutative_square:
     assumes "commutative_square f g h k" and "seq l f"
     shows "commutative_square (l \<cdot> f) (l \<cdot> g) h k"
       using assms comp_assoc
       by (elim commutative_squareE, intro commutative_squareI, auto)
 
   end
 
   section "Cospan Diagrams"
 
   (* TODO: Rework the ugly development of equalizers into this form. *)
 
   text \<open>
     The ``shape'' of a cospan diagram is a category having two non-identity arrows
     with distinct domains and a common codomain.
   \<close>
 
   locale cospan_shape
   begin
 
     datatype Arr = Null | AA | BB | TT | AT | BT
 
     fun comp
     where "comp AA AA = AA"
         | "comp AT AA = AT"
         | "comp TT AT = AT"
         | "comp BB BB = BB"
         | "comp BT BB = BT"
         | "comp TT BT = BT"
         | "comp TT TT = TT"
         | "comp _ _ = Null"
 
     interpretation partial_magma comp
     proof
       show "\<exists>!n. \<forall>f. comp n f = n \<and> comp f n = n"
       proof
         show "\<forall>f. comp Null f = Null \<and> comp f Null = Null" by simp
         show "\<And>n. \<forall>f. comp n f = n \<and> comp f n = n \<Longrightarrow> n = Null"
           by (metis comp.simps(8))
       qed
     qed
 
     lemma null_char:
     shows "null = Null"
     proof -
       have "\<forall>f. comp Null f = Null \<and> comp f Null = Null" by simp
       thus ?thesis
         using null_def ex_un_null theI [of "\<lambda>n. \<forall>f. comp n f = n \<and> comp f n = n"]
         by (metis partial_magma.comp_null(2) partial_magma_axioms)
     qed
 
     lemma ide_char:
     shows "ide f \<longleftrightarrow> f = AA \<or> f = BB \<or> f = TT"
     proof
       show "ide f \<Longrightarrow> f = AA \<or> f = BB \<or> f = TT"
         using ide_def null_char by (cases f, simp_all)
       show "f = AA \<or> f = BB \<or> f = TT \<Longrightarrow> ide f"
       proof -
         have 1: "\<And>f g. f = AA \<or> f = BB \<or> f = TT \<Longrightarrow>
                        comp f f \<noteq> Null \<and>
                        (comp g f \<noteq> Null \<longrightarrow> comp g f = g) \<and>
                        (comp f g \<noteq> Null \<longrightarrow> comp f g = g)"
         proof -
           fix f g
           show "f = AA \<or> f = BB \<or> f = TT \<Longrightarrow>
                   comp f f \<noteq> Null \<and>
                   (comp g f \<noteq> Null \<longrightarrow> comp g f = g) \<and>
                   (comp f g \<noteq> Null \<longrightarrow> comp f g = g)"
             by (cases f; cases g, auto)
         qed
         assume f: "f = AA \<or> f = BB \<or> f = TT"
         show "ide f"
           using f 1 ide_def null_char by simp
       qed
     qed
 
     fun Dom
     where "Dom AA = AA"
         | "Dom BB = BB"
         | "Dom TT = TT"
         | "Dom AT = AA"
         | "Dom BT = BB"
         | "Dom _ = Null"
 
     fun Cod
     where "Cod AA = AA"
         | "Cod BB = BB"
         | "Cod TT = TT"
         | "Cod AT = TT"
         | "Cod BT = TT"
         | "Cod _ = Null"
 
     lemma domains_char':
     shows "domains f = (if f = Null then {} else {Dom f})"
       using domains_def ide_char null_char
       by (cases f, auto)
 
     lemma codomains_char':
     shows "codomains f = (if f = Null then {} else {Cod f})"
       using codomains_def ide_char null_char
       by (cases f, auto)
 
     lemma arr_char:
     shows "arr f \<longleftrightarrow> f \<noteq> Null"
       using arr_def domains_char' codomains_char' by simp
 
     lemma seq_char:
     shows "seq g f \<longleftrightarrow> (f = AA \<and> (g = AA \<or> g = AT)) \<or>
                        (f = BB \<and> (g = BB \<or> g = BT)) \<or>
                        (f = AT \<and> g = TT) \<or>
                        (f = BT \<and> g = TT) \<or>
                        (f = TT \<and> g = TT)"
       using arr_char null_char
       by (cases f; cases g, simp_all)
 
     interpretation category comp
     proof
       fix f g h
       show "comp g f \<noteq> null \<Longrightarrow> seq g f"
         using null_char arr_char seq_char by simp
       show "domains f \<noteq> {} \<longleftrightarrow> codomains f \<noteq> {}"
         using domains_char' codomains_char' by auto
       show "seq h g \<Longrightarrow> seq (comp h g) f \<Longrightarrow> seq g f"
         using seq_char arr_char
         by (cases g; cases h; simp_all)
       show "seq h (comp g f) \<Longrightarrow> seq g f \<Longrightarrow> seq h g"
         using seq_char arr_char
         by (cases f; cases g; simp_all)
       show "seq g f \<Longrightarrow> seq h g \<Longrightarrow> seq (comp h g) f"
         using seq_char arr_char
         by (cases f; simp_all; cases g; simp_all; cases h; auto)
       show "seq g f \<Longrightarrow> seq h g \<Longrightarrow> comp (comp h g) f = comp h (comp g f)"
        using seq_char
        by (cases f; simp_all; cases g; simp_all; cases h; auto)
     qed
 
     lemma is_category:
     shows "category comp"
       ..
 
     lemma dom_char:
     shows "dom = Dom"
       using dom_def domains_char domains_char' null_char by fastforce
 
     lemma cod_char:
     shows "cod = Cod"
       using cod_def codomains_char codomains_char' null_char by fastforce
 
   end
 
   sublocale cospan_shape \<subseteq> category comp
     using is_category by auto
 
   locale cospan_diagram =
     J: cospan_shape +
     C: category C
   for C :: "'c comp"      (infixr "\<cdot>" 55)
   and f0 :: 'c
   and f1 :: 'c +
   assumes is_cospan: "C.cospan f0 f1"
   begin
 
     no_notation J.comp   (infixr "\<cdot>" 55)
     notation J.comp      (infixr "\<cdot>\<^sub>J" 55)
 
     fun map
     where "map J.AA = C.dom f0"
         | "map J.BB = C.dom f1"
         | "map J.TT = C.cod f0"
         | "map J.AT = f0"
         | "map J.BT = f1"
         | "map _ = C.null"
 
   end
 
   sublocale cospan_diagram \<subseteq> diagram J.comp C map
   proof
     show "\<And>f. \<not> J.arr f \<Longrightarrow> map f = C.null"
       using J.arr_char by simp
     fix f
     assume f: "J.arr f"
     show "C.arr (map f)"
       using f J.arr_char is_cospan by (cases f, simp_all)
     show "C.dom (map f) = map (J.dom f)"
       using f J.arr_char J.dom_char is_cospan by (cases f, simp_all)
     show "C.cod (map f) = map (J.cod f)"
       using f J.arr_char J.cod_char is_cospan by (cases f, simp_all)
     next
     fix f g
     assume fg: "J.seq g f"
     show "map (g \<cdot>\<^sub>J f) = map g \<cdot> map f"
       using fg J.seq_char J.null_char J.not_arr_null is_cospan
       apply (cases f; cases g, simp_all)
       using C.comp_arr_dom C.comp_cod_arr by auto
   qed
 
   section "Category with Pullbacks"
 
   text \<open>
     A \emph{pullback} in a category @{term C} is a limit of a cospan diagram in @{term C}.
   \<close>
 
   context cospan_diagram
   begin
 
     definition mkCone
     where "mkCone p0 p1 \<equiv> \<lambda>j. if j = J.AA then p0
                                else if j = J.BB then p1
                                else if j = J.AT then f0 \<cdot> p0
                                else if j = J.BT then f1 \<cdot> p1
                                else if j = J.TT then f0 \<cdot> p0
                                else C.null"
 
     abbreviation is_rendered_commutative_by
     where "is_rendered_commutative_by p0 p1 \<equiv> C.seq f0 p0 \<and> f0 \<cdot> p0 = f1 \<cdot> p1"
 
     abbreviation has_as_pullback
     where "has_as_pullback p0 p1 \<equiv> limit_cone (C.dom p0) (mkCone p0 p1)"
 
     lemma cone_mkCone:
     assumes "is_rendered_commutative_by p0 p1"
     shows "cone (C.dom p0) (mkCone p0 p1)"
     proof -
       interpret E: constant_functor J.comp C \<open>C.dom p0\<close>
         apply unfold_locales using assms by auto
       show "cone (C.dom p0) (mkCone p0 p1)"
       proof
         fix f
         show "\<not> J.arr f \<Longrightarrow> mkCone p0 p1 f = C.null"
           using mkCone_def J.arr_char by simp
         assume f: "J.arr f"
         show "C.dom (mkCone p0 p1 f) = E.map (J.dom f)"
           using assms f mkCone_def J.arr_char J.dom_char
           apply (cases f, simp_all)
           by (metis C.dom_comp)+
         show "C.cod (mkCone p0 p1 f) = map (J.cod f)"
           using assms f mkCone_def J.arr_char J.cod_char is_cospan
           by (cases f, auto)
         show "map f \<cdot> mkCone p0 p1 (J.dom f) = mkCone p0 p1 f"
           using assms f mkCone_def J.arr_char J.dom_char C.comp_ide_arr is_cospan
           by (cases f, auto)
         show "mkCone p0 p1 (J.cod f) \<cdot> E.map f = mkCone p0 p1 f"
           using assms f mkCone_def J.arr_char J.cod_char C.comp_arr_dom
           apply (cases f, auto)
              apply (metis C.dom_comp C.seqE)
           by (metis C.dom_comp)+
       qed
     qed
 
     lemma is_rendered_commutative_by_cone:
     assumes "cone a \<chi>"
     shows "is_rendered_commutative_by (\<chi> J.AA) (\<chi> J.BB)"
     proof -
       interpret \<chi>: cone J.comp C map a \<chi>
         using assms by auto
       show ?thesis
       proof
         show "C.seq f0 (\<chi> J.AA)"
           by (metis C.seqI J.cod_char J.seq_char \<chi>.preserves_cod \<chi>.preserves_reflects_arr
               J.seqE is_cospan J.Cod.simps(1) map.simps(1))
         show "f0 \<cdot> \<chi> J.AA = f1 \<cdot> \<chi> J.BB"
           by (metis J.cod_char J.dom_char \<chi>.A.map_simp \<chi>.naturality
               J.Cod.simps(4-5) J.Dom.simps(4-5) J.comp.simps(2,5) J.seq_char map.simps(4-5))
       qed
     qed
 
     lemma mkCone_cone:
     assumes "cone a \<chi>"
     shows "mkCone (\<chi> J.AA) (\<chi> J.BB) = \<chi>"
     proof -
       interpret \<chi>: cone J.comp C map a \<chi>
         using assms by auto
       have 1: "is_rendered_commutative_by (\<chi> J.AA) (\<chi> J.BB)"
         using assms is_rendered_commutative_by_cone by blast
       interpret mkCone_\<chi>: cone J.comp C map \<open>C.dom (\<chi> J.AA)\<close> \<open>mkCone (\<chi> J.AA) (\<chi> J.BB)\<close>
         using assms cone_mkCone 1 by auto
       show ?thesis
       proof -
         have "\<And>j. j = J.AA \<Longrightarrow> mkCone (\<chi> J.AA) (\<chi> J.BB) j = \<chi> j"
           using mkCone_def \<chi>.is_extensional by simp
         moreover have "\<And>j. j = J.BB \<Longrightarrow> mkCone (\<chi> J.AA) (\<chi> J.BB) j = \<chi> j"
           using mkCone_def \<chi>.is_extensional by simp
         moreover have "\<And>j. j = J.TT \<Longrightarrow> mkCone (\<chi> J.AA) (\<chi> J.BB) j = \<chi> j"
           using 1 mkCone_def \<chi>.is_extensional \<chi>.A.map_simp \<chi>.preserves_comp_1
                 cospan_shape.seq_char \<chi>.is_natural_2
           apply simp
           by (metis J.seqE J.comp.simps(5) map.simps(5))
         ultimately have "\<And>j. J.ide j \<Longrightarrow> mkCone (\<chi> J.AA) (\<chi> J.BB) j = \<chi> j"
           using J.ide_char by auto
         thus "mkCone (\<chi> J.AA) (\<chi> J.BB) = \<chi>"
           using mkCone_def NaturalTransformation.eqI [of J.comp C]
                 \<chi>.natural_transformation_axioms mkCone_\<chi>.natural_transformation_axioms
                 J.ide_char
           by simp
       qed
     qed
 
   end
 
   locale pullback_cone =
     J: cospan_shape +
     C: category C +
     D: cospan_diagram C f0 f1 +
     limit_cone J.comp C D.map \<open>C.dom p0\<close> \<open>D.mkCone p0 p1\<close>
   for C :: "'c comp"      (infixr "\<cdot>" 55)
   and f0 :: 'c
   and f1 :: 'c
   and p0 :: 'c
   and p1 :: 'c
   begin
 
     (* TODO: Equalizer should be simplifiable in the same way. *)
     lemma renders_commutative:
     shows "D.is_rendered_commutative_by p0 p1"
       using D.mkCone_def D.cospan_diagram_axioms cone_axioms
             cospan_diagram.is_rendered_commutative_by_cone
       by fastforce
 
     lemma is_universal':
     assumes "D.is_rendered_commutative_by p0' p1'"
     shows "\<exists>!h. \<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright> \<and> p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
     proof -
       have "D.cone (C.dom p0') (D.mkCone p0' p1')"
         using assms D.cone_mkCone by blast
       hence 1: "\<exists>!h. \<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright> \<and>
                      D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1'"
         using is_universal by simp
       have 2: "\<And>h. \<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright> \<Longrightarrow>
                     D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1' \<longleftrightarrow>
                     p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
       proof -
         fix h
         assume h: "\<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright>"
         show "D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1' \<longleftrightarrow>
               p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
         proof
           assume 3: "D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1'"
           show "p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
           proof
             show "p0 \<cdot> h = p0'"
             proof -
               have "p0' = D.mkCone p0' p1' J.AA"
                 using D.mkCone_def J.arr_char by simp
               also have "... = D.cones_map h (D.mkCone p0 p1) J.AA"
                 using 3 by simp
               also have "... = p0 \<cdot> h"
                 using h D.mkCone_def J.arr_char cone_\<chi> by auto
               finally show ?thesis by auto
             qed
             show "p1 \<cdot> h = p1'"
             proof -
               have "p1' = D.mkCone p0' p1' J.BB"
                 using D.mkCone_def J.arr_char by simp
               also have "... = D.cones_map h (D.mkCone p0 p1) J.BB"
                 using 3 by simp
               also have "... = p1 \<cdot> h"
                 using h D.mkCone_def J.arr_char cone_\<chi> by auto
               finally show ?thesis by auto
             qed
           qed
           next
           assume 4: "p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
           show "D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1'"
           proof
             fix j
             have "\<not> J.arr j \<Longrightarrow> D.cones_map h (D.mkCone p0 p1) j = D.mkCone p0' p1' j"
               using h cone_axioms D.mkCone_def J.arr_char by auto
             moreover have "J.arr j \<Longrightarrow>
                            D.cones_map h (D.mkCone p0 p1) j = D.mkCone p0' p1' j"
               using assms h 4 cone_\<chi> D.mkCone_def J.arr_char renders_commutative
                     C.comp_assoc
               by fastforce
             ultimately show "D.cones_map h (D.mkCone p0 p1) j = D.mkCone p0' p1' j"
               using J.arr_char J.Dom.cases by blast
           qed
         qed
       qed
       thus ?thesis using 1 by blast
     qed
 
     lemma induced_arrowI':
     assumes "D.is_rendered_commutative_by p0' p1'"
     shows "\<guillemotleft>induced_arrow (C.dom p0') (D.mkCone p0' p1') : C.dom p0' \<rightarrow> C.dom p0\<guillemotright>"
     and "p0 \<cdot> induced_arrow (C.dom p0') (D.mkCone p0' p1') = p0'"
     and "p1 \<cdot> induced_arrow (C.dom p1') (D.mkCone p0' p1') = p1'"
     proof -
       interpret A': constant_functor J.comp C \<open>C.dom p0'\<close>
         using assms by (unfold_locales, auto)
       have cone: "D.cone (C.dom p0') (D.mkCone p0' p1')"
         using assms D.cone_mkCone [of p0' p1'] by blast
       show 1: "p0 \<cdot> induced_arrow (C.dom p0') (D.mkCone p0' p1') = p0'"
       proof -
         have "p0 \<cdot> induced_arrow (C.dom p0') (D.mkCone p0' p1') =
                 D.cones_map (induced_arrow (C.dom p0') (D.mkCone p0' p1'))
                             (D.mkCone p0 p1) J.AA"
           using cone induced_arrowI(1) D.mkCone_def J.arr_char cone_\<chi> by force
         also have "... = p0'"
         proof -
           have "D.cones_map (induced_arrow (C.dom p0') (D.mkCone p0' p1'))
                             (D.mkCone p0 p1) =
                 D.mkCone p0' p1'"
             using cone induced_arrowI by blast
           thus ?thesis
             using J.arr_char D.mkCone_def by simp
         qed
         finally show ?thesis by auto
       qed
       show 2: "p1 \<cdot> induced_arrow (C.dom p1') (D.mkCone p0' p1') = p1'"
       proof -
         have "p1 \<cdot> induced_arrow (C.dom p1') (D.mkCone p0' p1') =
               D.cones_map (induced_arrow (C.dom p0') (D.mkCone p0' p1'))
                           (D.mkCone p0 p1) J.BB"
         proof -
           have "C.dom p0' = C.dom p1'"
             using assms by (metis C.dom_comp)
           thus ?thesis
             using cone induced_arrowI(1) D.mkCone_def J.arr_char cone_\<chi> by force
         qed
         also have "... = p1'"
         proof -
           have "D.cones_map (induced_arrow (C.dom p0') (D.mkCone p0' p1'))
                             (D.mkCone p0 p1) =
                 D.mkCone p0' p1'"
             using cone induced_arrowI by blast
           thus ?thesis
             using J.arr_char D.mkCone_def by simp
         qed
         finally show ?thesis by auto
       qed
       show "\<guillemotleft>induced_arrow (C.dom p0') (D.mkCone p0' p1') : C.dom p0' \<rightarrow> C.dom p0\<guillemotright>"
         using 1 cone induced_arrowI by simp
     qed
 
   end
 
   context category
   begin
 
     definition has_as_pullback
     where "has_as_pullback f0 f1 p0 p1 \<equiv>
            cospan f0 f1 \<and> cospan_diagram.has_as_pullback C f0 f1 p0 p1"
 
     definition has_pullbacks
     where "has_pullbacks = (\<forall>f0 f1. cospan f0 f1 \<longrightarrow> (\<exists>p0 p1. has_as_pullback f0 f1 p0 p1))"
 
   end
 
   locale category_with_pullbacks =
     category +
   assumes has_pullbacks: has_pullbacks
 
   section "Elementary Category with Pullbacks"
 
   text \<open>
     An \emph{elementary category with pullbacks} is a category equipped with a specific
     way of mapping each cospan to a span such that the resulting square commutes and
     such that the span is universal for that property.  It is useful to assume that the
     functions, mapping a cospan to the two projections of the pullback, are extensional;
     that is, they yield @{term null} when applied to arguments that do not form a cospan.
   \<close>
 
   locale elementary_category_with_pullbacks =
     category C
   for C :: "'a comp"                              (infixr "\<cdot>" 55)
   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"                    ("\<p>\<^sub>0[_, _]")
   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"                    ("\<p>\<^sub>1[_, _]") +
   assumes prj0_ext: "\<not> cospan f g \<Longrightarrow> \<p>\<^sub>0[f, g] = null"
   and prj1_ext: "\<not> cospan f g \<Longrightarrow> \<p>\<^sub>1[f, g] = null"
   and pullback_commutes [intro]: "cospan f g \<Longrightarrow> commutative_square f g \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]"
   and universal: "commutative_square f g h k \<Longrightarrow> \<exists>!l. \<p>\<^sub>1[f, g] \<cdot> l = h \<and> \<p>\<^sub>0[f, g] \<cdot> l = k"
   begin
 
     lemma pullback_commutes':
     assumes "cospan f g"
     shows "f \<cdot> \<p>\<^sub>1[f, g] = g \<cdot> \<p>\<^sub>0[f, g]"
       using assms commutative_square_def by blast
 
     lemma prj0_in_hom':
     assumes "cospan f g"
     shows "\<guillemotleft>\<p>\<^sub>0[f, g] : dom \<p>\<^sub>0[f, g] \<rightarrow> dom g\<guillemotright>"
       using assms pullback_commutes
       by (metis category.commutative_squareE category_axioms in_homI)
 
     lemma prj1_in_hom':
     assumes "cospan f g"
     shows "\<guillemotleft>\<p>\<^sub>1[f, g] : dom \<p>\<^sub>0[f, g] \<rightarrow> dom f\<guillemotright>"
       using assms pullback_commutes
       by (metis category.commutative_squareE category_axioms in_homI)
 
     text \<open>
       The following gives us a notation for the common domain of the two projections
       of a pullback.
     \<close>
 
     definition pbdom        (infix "\<down>\<down>" 51)
     where "f \<down>\<down> g \<equiv> dom \<p>\<^sub>0[f, g]"
 
     lemma pbdom_in_hom [intro]:
     assumes "cospan f g"
     shows "\<guillemotleft>f \<down>\<down> g : f \<down>\<down> g \<rightarrow> f \<down>\<down> g\<guillemotright>"
       unfolding pbdom_def
       using assms prj0_in_hom'
       by (metis arr_dom_iff_arr arr_iff_in_hom cod_dom dom_dom in_homE)
 
     lemma ide_pbdom [simp]:
     assumes "cospan f g"
     shows "ide (f \<down>\<down> g)"
       using assms ide_in_hom by auto[1]
 
     lemma prj0_in_hom [intro, simp]:
     assumes "cospan f g" and "a = f \<down>\<down> g" and "b = dom g"
     shows "\<guillemotleft>\<p>\<^sub>0[f, g] : a \<rightarrow> b\<guillemotright>"
       unfolding pbdom_def
       using assms prj0_in_hom' by (simp add: pbdom_def)
 
     lemma prj1_in_hom [intro, simp]:
     assumes "cospan f g" and "a = f \<down>\<down> g" and "b = dom f"
     shows "\<guillemotleft>\<p>\<^sub>1[f, g] : a \<rightarrow> b\<guillemotright>"
       unfolding pbdom_def
       using assms prj1_in_hom' by (simp add: pbdom_def)
 
     lemma prj0_simps [simp]:
     assumes "cospan f g"
     shows "arr \<p>\<^sub>0[f, g]" and "dom \<p>\<^sub>0[f, g] = f \<down>\<down> g" and "cod \<p>\<^sub>0[f, g] = dom g"
       using assms prj0_in_hom by (blast, blast, blast)
 
     lemma prj0_simps_arr [iff]:
     shows "arr \<p>\<^sub>0[f, g] \<longleftrightarrow> cospan f g"
     proof
       show "cospan f g \<Longrightarrow> arr \<p>\<^sub>0[f, g]"
         using prj0_in_hom by auto
       show "arr \<p>\<^sub>0[f, g] \<Longrightarrow> cospan f g"
         using prj0_ext not_arr_null by metis
     qed
 
     lemma prj1_simps [simp]:
     assumes "cospan f g"
     shows "arr \<p>\<^sub>1[f, g]" and "dom \<p>\<^sub>1[f, g] = f \<down>\<down> g" and "cod \<p>\<^sub>1[f, g] = dom f"
       using assms prj1_in_hom by (blast, blast, blast)
 
     lemma prj1_simps_arr [iff]:
     shows "arr \<p>\<^sub>1[f, g] \<longleftrightarrow> cospan f g"
     proof
       show "cospan f g \<Longrightarrow> arr \<p>\<^sub>1[f, g]"
         using prj1_in_hom by auto
       show "arr \<p>\<^sub>1[f, g] \<Longrightarrow> cospan f g"
         using prj1_ext not_arr_null by metis
     qed
 
     lemma span_prj:
     assumes "cospan f g"
     shows "span \<p>\<^sub>0[f, g] \<p>\<^sub>1[f, g]"
       using assms by simp
 
     text \<open>
       We introduce a notation for tupling, which produces the induced arrow into a pullback.
       In our notation, the ``$0$-side'', which we regard as the input, occurs on the right,
       and the ``$1$-side'', which we regard as the output, occurs on the left.
     \<close>
 
     definition tuple         ("\<langle>_ \<lbrakk>_, _\<rbrakk> _\<rangle>")
     where "\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<equiv> if commutative_square f g h k then
                            THE l. \<p>\<^sub>0[f, g] \<cdot> l = k \<and> \<p>\<^sub>1[f, g] \<cdot> l = h
                          else null"
 
     lemma tuple_in_hom [intro]:
     assumes "commutative_square f g h k"
     shows "\<guillemotleft>\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> : dom h \<rightarrow> f \<down>\<down> g\<guillemotright>"
     proof
       have 1: "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = k \<and> \<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = h"
         unfolding tuple_def
         using assms universal theI [of "\<lambda>l. \<p>\<^sub>0[f, g] \<cdot> l = k \<and> \<p>\<^sub>1[f, g] \<cdot> l = h"]
         apply simp
         by meson
       show "arr \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle>"
         using assms 1
         apply (elim commutative_squareE)
         by (metis (no_types, lifting) seqE)
       show "dom \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = dom h"
         using assms 1
         apply (elim commutative_squareE)
         by (metis (no_types, lifting) dom_comp)
       show "cod \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = f \<down>\<down> g"
         unfolding pbdom_def
         using assms 1
         apply (elim commutative_squareE)
         by (metis seqE)
     qed
 
     lemma tuple_is_extensional:
     assumes "\<not> commutative_square f g h k"
     shows "\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = null"
       unfolding tuple_def
       using assms by simp
 
     lemma tuple_simps [simp]:
     assumes "commutative_square f g h k"
     shows "arr \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle>" and "dom \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = dom h" and "cod \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = f \<down>\<down> g"
       using assms tuple_in_hom apply blast
       using assms tuple_in_hom apply blast
       using assms tuple_in_hom by blast
 
     lemma prj_tuple [simp]:
     assumes "commutative_square f g h k"
     shows "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = k" and "\<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = h"
     proof -
       have 1: "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = k \<and> \<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = h"
         unfolding tuple_def
         using assms universal theI [of "\<lambda>l. \<p>\<^sub>0[f, g] \<cdot> l = k \<and> \<p>\<^sub>1[f, g] \<cdot> l = h"]
         apply simp
         by meson
       show "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = k" using 1 by simp
       show "\<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = h" using 1 by simp
     qed
 
     lemma tuple_prj:
     assumes "cospan f g" and "seq \<p>\<^sub>1[f, g] h"
     shows "\<langle>\<p>\<^sub>1[f, g] \<cdot> h \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g] \<cdot> h\<rangle> = h"
     proof -
       have 1: "commutative_square f g (\<p>\<^sub>1[f, g] \<cdot> h) (\<p>\<^sub>0[f, g] \<cdot> h)"
         using assms pullback_commutes
         by (simp add: commutative_square_comp_arr)
       have "\<p>\<^sub>0[f, g] \<cdot> \<langle>\<p>\<^sub>1[f, g] \<cdot> h \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g] \<cdot> h\<rangle> = \<p>\<^sub>0[f, g] \<cdot> h"
         using assms 1 by simp
       moreover have "\<p>\<^sub>1[f, g] \<cdot> \<langle>\<p>\<^sub>1[f, g] \<cdot> h \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g] \<cdot> h\<rangle> = \<p>\<^sub>1[f, g] \<cdot> h"
         using assms 1 by simp
       ultimately show ?thesis
         unfolding tuple_def
         using assms 1 universal [of f g "\<p>\<^sub>1[f, g] \<cdot> h" "\<p>\<^sub>0[f, g] \<cdot> h"]
               theI_unique [of "\<lambda>l. \<p>\<^sub>0[f, g] \<cdot> l = \<p>\<^sub>0[f, g] \<cdot> h \<and> \<p>\<^sub>1[f, g] \<cdot> l = \<p>\<^sub>1[f, g] \<cdot> h" h]
         by auto
     qed
 
     lemma tuple_prj_spc [simp]:
     assumes "cospan f g"
     shows "\<langle>\<p>\<^sub>1[f, g] \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g]\<rangle> = f \<down>\<down> g"
     proof -
       have "\<langle>\<p>\<^sub>1[f, g] \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g]\<rangle> = \<langle>\<p>\<^sub>1[f, g] \<cdot> (f \<down>\<down> g) \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g] \<cdot> (f \<down>\<down> g)\<rangle>"
         using assms comp_arr_dom by simp
       thus ?thesis
         using assms tuple_prj by simp
     qed
 
     lemma prj_joint_monic:
     assumes "cospan f g" and "seq \<p>\<^sub>1[f, g] h" and "seq \<p>\<^sub>1[f, g] h'"
     and "\<p>\<^sub>0[f, g] \<cdot> h = \<p>\<^sub>0[f, g] \<cdot> h'" and "\<p>\<^sub>1[f, g] \<cdot> h = \<p>\<^sub>1[f, g] \<cdot> h'"
     shows "h = h'"
     proof -
       have "h = \<langle>\<p>\<^sub>1[f, g] \<cdot> h \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g] \<cdot> h\<rangle>"
         using assms tuple_prj [of f g h] by simp
       also have "... = \<langle>\<p>\<^sub>1[f, g] \<cdot> h' \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g] \<cdot> h'\<rangle>"
         using assms by simp
       also have "... = h'"
         using assms tuple_prj [of f g h'] by simp
       finally show ?thesis by blast
     qed
 
     text \<open>
       The pullback of an identity along an arbitrary arrow is an isomorphism.
     \<close>
 
     lemma iso_pullback_ide:
     assumes "cospan \<mu> \<nu>" and "ide \<mu>"
     shows "iso \<p>\<^sub>0[\<mu>, \<nu>]"
     proof -
       have "inverse_arrows \<p>\<^sub>0[\<mu>, \<nu>] \<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle>"
       proof
         show 1: "ide (\<p>\<^sub>0[\<mu>, \<nu>] \<cdot> \<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle>)"
           using assms comp_arr_dom comp_cod_arr prj_tuple(1) by simp
         show "ide (\<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle> \<cdot> \<p>\<^sub>0[\<mu>, \<nu>])"
         proof -
           have "\<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle> \<cdot> \<p>\<^sub>0[\<mu>, \<nu>] = (\<mu> \<down>\<down> \<nu>)"
           proof -
             have "\<p>\<^sub>0[\<mu>, \<nu>] \<cdot> \<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle> \<cdot> \<p>\<^sub>0[\<mu>, \<nu>] = \<p>\<^sub>0[\<mu>, \<nu>] \<cdot> (\<mu> \<down>\<down> \<nu>)"
             proof -
               have "\<p>\<^sub>0[\<mu>, \<nu>] \<cdot> \<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle> \<cdot> \<p>\<^sub>0[\<mu>, \<nu>] = (\<p>\<^sub>0[\<mu>, \<nu>] \<cdot> \<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle>) \<cdot> \<p>\<^sub>0[\<mu>, \<nu>]"
                 using assms 1 comp_reduce by blast
               also have "... = \<p>\<^sub>0[\<mu>, \<nu>] \<cdot> (\<mu> \<down>\<down> \<nu>)"
                 using assms prj_tuple(1) pullback_commutes comp_arr_dom comp_cod_arr by simp
               finally show ?thesis by blast
             qed
             moreover have "\<p>\<^sub>1[\<mu>, \<nu>] \<cdot> \<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle> \<cdot> \<p>\<^sub>0[\<mu>, \<nu>] = \<p>\<^sub>1[\<mu>, \<nu>] \<cdot> (\<mu> \<down>\<down> \<nu>)"
             proof -
               have "\<p>\<^sub>1[\<mu>, \<nu>] \<cdot> \<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle> \<cdot> \<p>\<^sub>0[\<mu>, \<nu>] = (\<p>\<^sub>1[\<mu>, \<nu>] \<cdot> \<langle>\<nu> \<lbrakk>\<mu>, \<nu>\<rbrakk> dom \<nu>\<rangle>) \<cdot> \<p>\<^sub>0[\<mu>, \<nu>]"
                 using assms(2) comp_assoc by simp
               also have "... = \<nu> \<cdot> \<p>\<^sub>0[\<mu>, \<nu>]"
                 using assms comp_arr_dom comp_cod_arr prj_tuple(2) by fastforce
               also have "... = \<mu> \<cdot> \<p>\<^sub>1[\<mu>, \<nu>]"
                 using assms pullback_commutes commutative_square_def by simp
               also have "... = \<p>\<^sub>1[\<mu>, \<nu>] \<cdot> (\<mu> \<down>\<down> \<nu>)"
                 using assms comp_arr_dom comp_cod_arr pullback_commutes commutative_square_def
                 by simp
               finally show ?thesis by simp
             qed
             ultimately show ?thesis
               using assms prj0_in_hom prj1_in_hom comp_arr_dom prj1_simps(1-2) prj_joint_monic
               by metis
           qed
           thus ?thesis
             using assms by auto
         qed
       qed
       thus ?thesis by auto
     qed
 
     lemma comp_tuple_arr:
     assumes "commutative_square f g h k" and "seq h l"
     shows "\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<cdot> l = \<langle>h \<cdot> l \<lbrakk>f, g\<rbrakk> k \<cdot> l\<rangle>"
     proof -
       have "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<cdot> l = \<p>\<^sub>0[f, g] \<cdot> \<langle>h \<cdot> l \<lbrakk>f, g\<rbrakk> k \<cdot> l\<rangle>"
       proof -
         have "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<cdot> l = (\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle>) \<cdot> l"
           using comp_assoc by simp
         also have "... = \<p>\<^sub>0[f, g] \<cdot> \<langle>h \<cdot> l \<lbrakk>f, g\<rbrakk> k \<cdot> l\<rangle>"
           using assms commutative_square_comp_arr by auto
         finally show ?thesis by blast
       qed
       moreover have "\<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<cdot> l = \<p>\<^sub>1[f, g] \<cdot> \<langle>h \<cdot> l \<lbrakk>f, g\<rbrakk> k \<cdot> l\<rangle>"
       proof -
         have "\<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<cdot> l = (\<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle>) \<cdot> l"
           using comp_assoc by simp
         also have "... = \<p>\<^sub>1[f, g] \<cdot> \<langle>h \<cdot> l \<lbrakk>f, g\<rbrakk> k \<cdot> l\<rangle>"
           using assms commutative_square_comp_arr by auto
         finally show ?thesis by blast
       qed
       moreover have "seq \<p>\<^sub>1[f, g] (\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<cdot> l)"
         using assms tuple_in_hom prj1_in_hom by fastforce
       ultimately show ?thesis
         using assms prj_joint_monic [of f g "\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<cdot> l" "\<langle>h \<cdot> l \<lbrakk>f, g\<rbrakk> k \<cdot> l\<rangle>"]
         by auto
     qed
 
     lemma pullback_arr_cod:
     assumes "arr f"
     shows "inverse_arrows \<p>\<^sub>1[f, cod f] \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle>"
     and "inverse_arrows \<p>\<^sub>0[cod f, f] \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle>"
     proof -
       show "inverse_arrows \<p>\<^sub>1[f, cod f] \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle>"
       proof
         show "ide (\<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle> \<cdot> \<p>\<^sub>1[f, cod f])"
         proof -
           have "\<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle> \<cdot> \<p>\<^sub>1[f, cod f] = f \<down>\<down> cod f"
           proof -
             have "\<p>\<^sub>0[f, cod f] \<cdot> \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle> \<cdot> \<p>\<^sub>1[f, cod f] = \<p>\<^sub>0[f, cod f] \<cdot> (f \<down>\<down> cod f)"
             proof -
               have "\<p>\<^sub>0[f, cod f] \<cdot> \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle> \<cdot> \<p>\<^sub>1[f, cod f] =
                     (\<p>\<^sub>0[f, cod f] \<cdot> \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle>) \<cdot> \<p>\<^sub>1[f, cod f]"
                 using comp_assoc by simp
               also have "... = \<p>\<^sub>0[f, cod f] \<cdot> (f \<down>\<down> cod f)"
                 using assms pullback_commutes [of f "cod f"] comp_arr_dom comp_cod_arr
                 by auto
               finally show ?thesis by blast
             qed
             moreover
             have "\<p>\<^sub>1[f, cod f] \<cdot> \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle> \<cdot> \<p>\<^sub>1[f, cod f] = \<p>\<^sub>1[f, cod f] \<cdot> (f \<down>\<down> cod f)"
             proof -
               have "\<p>\<^sub>1[f, cod f] \<cdot> \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle> \<cdot> \<p>\<^sub>1[f, cod f] =
                     (\<p>\<^sub>1[f, cod f] \<cdot> \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle>) \<cdot> \<p>\<^sub>1[f, cod f]"
                 using assms comp_assoc by presburger
               also have "... = \<p>\<^sub>1[f, cod f] \<cdot> (f \<down>\<down> cod f)"
                 using assms comp_arr_dom comp_cod_arr by simp
               finally show ?thesis by blast
             qed
             ultimately show ?thesis
               using assms
                     prj_joint_monic
                       [of f "cod f" "\<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle> \<cdot> \<p>\<^sub>1[f, cod f]" "f \<down>\<down> cod f"]
               by simp
           qed
           thus ?thesis
             using assms arr_cod cod_cod prj1_simps_arr by simp
         qed
         show "ide (\<p>\<^sub>1[f, cod f] \<cdot> \<langle>dom f \<lbrakk>f, cod f\<rbrakk> f\<rangle>)"
           using assms comp_arr_dom comp_cod_arr by fastforce
       qed
       show "inverse_arrows \<p>\<^sub>0[cod f, f] \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle>"
       proof
         show "ide (\<p>\<^sub>0[cod f, f] \<cdot> \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle>)"
           using assms comp_arr_dom comp_cod_arr by simp
         show "ide (\<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle> \<cdot> \<p>\<^sub>0[cod f, f])"
         proof -
           have "\<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle> \<cdot> \<p>\<^sub>0[cod f, f] = cod f \<down>\<down> f"
           proof -
             have "\<p>\<^sub>0[cod f, f] \<cdot> \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle> \<cdot> \<p>\<^sub>0[cod f, f] = \<p>\<^sub>0[cod f, f] \<cdot> (cod f \<down>\<down> f)"
             proof -
               have "\<p>\<^sub>0[cod f, f] \<cdot> \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle> \<cdot> \<p>\<^sub>0[cod f, f] =
                     (\<p>\<^sub>0[cod f, f] \<cdot> \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle>) \<cdot> \<p>\<^sub>0[cod f, f]"
                 using comp_assoc by simp
               also have "... = dom f \<cdot> \<p>\<^sub>0[cod f, f]"
                 using assms comp_arr_dom comp_cod_arr by simp
               also have "... = \<p>\<^sub>0[cod f, f] \<cdot> (cod f \<down>\<down> f)"
                 using assms comp_arr_dom comp_cod_arr by simp
               finally show ?thesis
                 using prj0_in_hom by blast
             qed
             moreover
             have "\<p>\<^sub>1[cod f, f] \<cdot> \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle> \<cdot> \<p>\<^sub>0[cod f, f] = \<p>\<^sub>1[cod f, f] \<cdot> (cod f \<down>\<down> f)"
             proof -
               have "\<p>\<^sub>1[cod f, f] \<cdot> \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle> \<cdot> \<p>\<^sub>0[cod f, f] =
                     (\<p>\<^sub>1[cod f, f] \<cdot> \<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle>) \<cdot> \<p>\<^sub>0[cod f, f]"
                 using comp_assoc by simp
               also have "... = \<p>\<^sub>1[cod f, f] \<cdot> (cod f \<down>\<down> f)"
                 using assms pullback_commutes [of "cod f" f] comp_arr_dom comp_cod_arr
                 by auto
               finally show ?thesis by blast
             qed
             ultimately show ?thesis
               using assms prj_joint_monic [of "cod f" f "\<langle>f \<lbrakk>cod f, f\<rbrakk> dom f\<rangle> \<cdot> \<p>\<^sub>0[cod f, f]"]
               by simp
           qed
           thus ?thesis using assms by simp
         qed
       qed
     qed
 
     text \<open>
       The pullback of a monomorphism along itself is automatically symmetric: the left
       and right projections are equal.
     \<close>
 
     lemma pullback_mono_self:
     assumes "mono f"
     shows "\<p>\<^sub>0[f, f] = \<p>\<^sub>1[f, f]"
     proof -
       have "f \<cdot> \<p>\<^sub>0[f, f] = f \<cdot> \<p>\<^sub>1[f, f]"
         using assms pullback_commutes [of f f]
         by (metis commutative_squareE mono_implies_arr)
       thus ?thesis
         using assms monoE [of f "\<p>\<^sub>1[f, f]" "\<p>\<^sub>0[f, f]"]
         by (metis mono_implies_arr prj0_simps(1,3) seqI)
     qed
 
     lemma pullback_iso_self:
     assumes "iso f"
     shows "\<p>\<^sub>0[f, f] = \<p>\<^sub>1[f, f]"
       using assms pullback_mono_self iso_is_section section_is_mono by simp
 
     lemma pullback_ide_self [simp]:
     assumes "ide a"
     shows "\<p>\<^sub>0[a, a] = \<p>\<^sub>1[a, a]"
       using assms pullback_iso_self ide_is_iso by blast
 
   end
 
   section "Agreement between the Definitions"
 
   text \<open>
     It is very easy to write locale assumptions that have unintended consequences
     or that are even inconsistent.  So, to keep ourselves honest, we don't just accept the
     definition of ``elementary category with pullbacks'', but in fact we formally establish
     the sense in which it agrees with our standard definition of ``category with pullbacks'',
     which is given in terms of limit cones.
     This is extra work, but it ensures that we didn't make a mistake.
   \<close>
 
   context category_with_pullbacks
   begin
 
     definition prj1
     where "prj1 f g \<equiv> if cospan f g then
                          fst (SOME x. cospan_diagram.has_as_pullback C f g (fst x) (snd x))
                        else null"
 
     definition prj0
     where "prj0 f g \<equiv> if cospan f g then
                          snd (SOME x. cospan_diagram.has_as_pullback C f g (fst x) (snd x))
                        else null"
 
     lemma prj_yields_pullback:
     assumes "cospan f g"
     shows "cospan_diagram.has_as_pullback C f g (prj1 f g) (prj0 f g)"
     proof -
       have "\<exists>x. cospan_diagram.has_as_pullback C f g (fst x) (snd x)"
         using assms has_pullbacks has_pullbacks_def has_as_pullback_def by simp
       thus ?thesis
         using assms has_pullbacks has_pullbacks_def prj0_def prj1_def
               someI_ex [of "\<lambda>x. cospan_diagram.has_as_pullback C f g (fst x) (snd x)"]
         by simp
     qed
 
     interpretation elementary_category_with_pullbacks C prj0 prj1
     proof
       show "\<And>f g. \<not> cospan f g \<Longrightarrow> prj0 f g = null"
         using prj0_def by auto
       show "\<And>f g. \<not> cospan f g \<Longrightarrow> prj1 f g = null"
         using prj1_def by auto
       show "\<And>f g. cospan f g \<Longrightarrow> commutative_square f g (prj1 f g) (prj0 f g)"
       proof
         fix f g
         assume fg: "cospan f g"
         show "cospan f g" by fact
         interpret J: cospan_shape .
         interpret D: cospan_diagram C f g
           using fg by (unfold_locales, auto)
         let ?\<chi> = "D.mkCone (prj1 f g) (prj0 f g)"
         interpret \<chi>: limit_cone J.comp C D.map \<open>dom (prj1 f g)\<close> ?\<chi>
           using fg prj_yields_pullback by auto
         have 1: "prj1 f g = ?\<chi> J.AA \<and> prj0 f g = ?\<chi> J.BB"
           using D.mkCone_def by simp
         show "span (prj1 f g) (prj0 f g)"
         proof -
           have "arr (prj1 f g) \<and> arr (prj0 f g)"
             using 1 J.arr_char J.seq_char
             by (metis J.seqE \<chi>.preserves_reflects_arr)
           moreover have "dom (prj1 f g) = dom (prj0 f g)"
             using 1 D.is_rendered_commutative_by_cone \<chi>.cone_axioms J.seq_char
             by (metis J.cod_eqI J.seqE \<chi>.A.map_simp \<chi>.preserves_dom J.ide_char)
           ultimately show ?thesis by simp
         qed
         show "dom f = cod (prj1 f g)"
           using 1 fg \<chi>.preserves_cod [of J.BB] J.cod_char D.mkCone_def
           by (metis D.map.simps(1) D.preserves_cod J.seqE \<chi>.preserves_cod cod_dom J.seq_char)
         show "f \<cdot> prj1 f g = g \<cdot> prj0 f g"
           using 1 fg D.is_rendered_commutative_by_cone \<chi>.cone_axioms by force
       qed
       show "\<And>f g h k. commutative_square f g h k \<Longrightarrow> \<exists>!l. prj1 f g \<cdot> l = h \<and> prj0 f g \<cdot> l = k"
       proof -
         fix f g h k
         assume fghk: "commutative_square f g h k"
         interpret J: cospan_shape .
         interpret D: cospan_diagram C f g
           using fghk by (unfold_locales, auto)
         let ?\<chi> = "D.mkCone (prj1 f g) (prj0 f g)"
         interpret \<chi>: limit_cone J.comp C D.map \<open>dom (prj1 f g)\<close> ?\<chi>
           using fghk prj_yields_pullback by auto
         interpret \<chi>: pullback_cone C f g \<open>prj1 f g\<close> \<open>prj0 f g\<close> ..
         have 1: "prj1 f g = ?\<chi> J.AA \<and> prj0 f g = ?\<chi> J.BB"
           using D.mkCone_def by simp
         show "\<exists>!l. prj1 f g \<cdot> l = h \<and> prj0 f g \<cdot> l = k"
         proof
           let ?l = "SOME l. prj1 f g \<cdot> l = h \<and> prj0 f g \<cdot> l = k"
           show "prj1 f g \<cdot> ?l = h \<and> prj0 f g \<cdot> ?l = k"
              using fghk \<chi>.is_universal' \<chi>.renders_commutative
                    someI_ex [of "\<lambda>l. prj1 f g \<cdot> l = h \<and> prj0 f g \<cdot> l = k"]
              by blast
           thus "\<And>l. prj1 f g \<cdot> l = h \<and> prj0 f g \<cdot> l = k \<Longrightarrow> l = ?l"
             using fghk \<chi>.is_universal' \<chi>.renders_commutative limit_cone_def
             by (metis (no_types, lifting) in_homI seqE commutative_squareE dom_comp seqI)
         qed
       qed
     qed
 
     proposition extends_to_elementary_category_with_pullbacks:
     shows "elementary_category_with_pullbacks C prj0 prj1"
       ..
 
   end
 
   context elementary_category_with_pullbacks
   begin
 
     interpretation category_with_pullbacks C
     proof
       show "has_pullbacks"
       proof (unfold has_pullbacks_def)
         have "\<And>f g. cospan f g \<Longrightarrow> \<exists>p0 p1. has_as_pullback f g p0 p1"
         proof -
           fix f g
           assume fg: "cospan f g"
           interpret J: cospan_shape .
           interpret D: cospan_diagram C f g
             using fg by (unfold_locales, auto)
           let ?\<chi> = "D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]"
           interpret \<chi>: cone J.comp C D.map \<open>dom \<p>\<^sub>1[f, g]\<close> ?\<chi>
           proof -
             have "D.is_rendered_commutative_by \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]"
               using fg pullback_commutes' by simp
             thus "cone J.comp C D.map (dom \<p>\<^sub>1[f, g]) ?\<chi>"
               using D.cone_mkCone by auto
           qed
           interpret \<chi>: limit_cone J.comp C D.map \<open>dom \<p>\<^sub>1[f, g]\<close> ?\<chi>
           proof
             fix a' \<chi>'
             assume \<chi>': "D.cone a' \<chi>'"
             interpret \<chi>': cone J.comp C D.map a' \<chi>'
               using \<chi>' by simp
             have 1: "commutative_square f g (\<chi>' J.AA) (\<chi>' J.BB)"
               using fg J.ide_char J.cod_char D.is_rendered_commutative_by_cone \<chi>'.cone_axioms
               by auto
             show "\<exists>!h. \<guillemotleft>h : a' \<rightarrow> dom \<p>\<^sub>1[f, g]\<guillemotright> \<and>
                        D.cones_map h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]) = \<chi>'"
             proof
               let ?h = "\<langle>\<chi>' J.AA \<lbrakk>f, g\<rbrakk> \<chi>' J.BB\<rangle>"
               show h': "\<guillemotleft>?h : a' \<rightarrow> dom \<p>\<^sub>1[f, g]\<guillemotright> \<and>
                         D.cones_map ?h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]) = \<chi>'"
               proof
                 show h: "\<guillemotleft>?h : a' \<rightarrow> dom \<p>\<^sub>1[f, g]\<guillemotright>"
                   using fg 1 by fastforce
                 show "D.cones_map ?h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]) = \<chi>'"
                 proof -
                   have "D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g] \<in> D.cones (cod \<langle>\<chi>' J.AA \<lbrakk>f, g\<rbrakk> \<chi>' J.BB\<rangle>)"
                     using fg h D.cone_mkCone D.is_rendered_commutative_by_cone
                           \<chi>.cone_axioms
                     by auto
                   hence 2: "D.cones_map ?h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]) \<in> D.cones a'"
                     using fg h D.cones_map_mapsto by blast
                   interpret \<chi>'h: cone J.comp C D.map a'
                                    \<open>D.cones_map ?h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g])\<close>
                     using 2 by simp
                   show ?thesis
                   proof -
                     have "\<And>j. J.ide j \<Longrightarrow> D.cones_map ?h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]) j = \<chi>' j"
                     proof -
                       fix j
                       show "J.ide j \<Longrightarrow> D.cones_map ?h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]) j = \<chi>' j"
                         using fg h 1 J.ide_char D.mkCone_def \<chi>.cone_axioms
                         apply (cases j, simp_all)
                         by (metis D.map.simps(4) J.dom_eqI \<chi>'.is_natural_1 \<chi>'.naturality
                             J.seqE \<chi>'.A.map_simp J.comp.simps(3,7) J.seq_char
                             prj_tuple(2) comp_assoc)
                     qed
                     thus ?thesis
                       using NaturalTransformation.eqI
                             \<chi>'.natural_transformation_axioms \<chi>'h.natural_transformation_axioms
                       by blast
                   qed
                 qed
               qed
               show "\<And>h. \<guillemotleft>h : a' \<rightarrow> dom \<p>\<^sub>1[f, g]\<guillemotright> \<and>
                         D.cones_map h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]) = \<chi>' \<Longrightarrow>
                           h = ?h"
               proof -
                 fix h
                 assume 2: "\<guillemotleft>h : a' \<rightarrow> dom \<p>\<^sub>1[f, g]\<guillemotright> \<and>
                         D.cones_map h (D.mkCone \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]) = \<chi>'"
                 show "h = ?h"
                 proof -
                   have "\<p>\<^sub>0[f, g] \<cdot> h = \<p>\<^sub>0[f, g] \<cdot> ?h \<and> \<p>\<^sub>1[f, g] \<cdot> h = \<p>\<^sub>1[f, g] \<cdot> ?h"
                     using 1 2 fg J.arr_char \<chi>.cone_axioms D.mkCone_def by auto
                   thus ?thesis
                     using fg 2 h' prj_joint_monic by blast
                 qed
               qed
             qed
           qed
           show "\<exists>p0 p1. has_as_pullback f g p0 p1"
             using fg has_as_pullback_def \<chi>.limit_cone_axioms by blast
         qed
         thus "\<forall>f g. cospan f g \<longrightarrow> (\<exists>p0 p1. has_as_pullback f g p0 p1)"
           by simp
       qed
     qed
 
     proposition is_category_with_pullbacks:
     shows "category_with_pullbacks C"
       ..
 
   end
 
   sublocale elementary_category_with_pullbacks \<subseteq> category_with_pullbacks
     using is_category_with_pullbacks by auto
 
 end
 
diff --git a/thys/MonoidalCategory/CartesianMonoidalCategory.thy b/thys/MonoidalCategory/CartesianMonoidalCategory.thy
--- a/thys/MonoidalCategory/CartesianMonoidalCategory.thy
+++ b/thys/MonoidalCategory/CartesianMonoidalCategory.thy
@@ -1,654 +1,654 @@
 (*  Title:       CartesianMonoidalCategory
     Author:      Eugene W. Stark <stark@cs.stonybrook.edu>, 2020
     Maintainer:  Eugene W. Stark <stark@cs.stonybrook.edu>
 *)
 
 chapter "Cartesian Monoidal Category"
 
 theory CartesianMonoidalCategory
-imports MonoidalCategory.MonoidalCategory Category3.CartesianCategory
+imports MonoidalCategory Category3.CartesianCategory
 begin
 
   locale symmetric_monoidal_category =
     monoidal_category C T \<alpha> \<iota> +
     S: symmetry_functor C C +
     ToS: composite_functor CC.comp CC.comp C S.map T +
     \<sigma>: natural_isomorphism CC.comp C T ToS.map \<sigma>
   for C :: "'a comp"                            (infixr "\<cdot>" 55)
   and T :: "'a * 'a \<Rightarrow> 'a"
   and \<alpha> :: "'a * 'a * 'a \<Rightarrow> 'a"
   and \<iota> :: 'a
   and \<sigma> :: "'a * 'a \<Rightarrow> 'a" +
   assumes sym_inverse: "\<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> inverse_arrows (\<sigma> (a, b)) (\<sigma> (b, a))"
   and unitor_coherence: "ide a \<Longrightarrow> \<l>[a] \<cdot> \<sigma> (a, \<I>) = \<r>[a]"
   and assoc_coherence: "\<lbrakk> ide a; ide b; ide c \<rbrakk> \<Longrightarrow>
                           \<alpha> (b, c, a) \<cdot> \<sigma> (a, b \<otimes> c) \<cdot> \<alpha> (a, b, c)
                              = (b \<otimes> \<sigma> (a, c)) \<cdot> \<alpha> (b, a, c) \<cdot> (\<sigma> (a, b) \<otimes> c)"
   begin
 
     abbreviation sym                  ("\<s>[_, _]")
     where "sym a b \<equiv> \<sigma> (a, b)"
 
   end
 
   locale elementary_symmetric_monoidal_category =
     elementary_monoidal_category C tensor unity lunit runit assoc
   for C :: "'a comp"                  (infixr "\<cdot>" 55)
   and tensor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"       (infixr "\<otimes>" 53)
   and unity :: 'a                      ("\<I>")
   and lunit :: "'a \<Rightarrow> 'a"              ("\<l>[_]")
   and runit :: "'a \<Rightarrow> 'a"              ("\<r>[_]")
   and assoc :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"  ("\<a>[_, _, _]")
   and sym :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"          ("\<s>[_, _]") +
   assumes sym_in_hom: "\<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> \<guillemotleft>\<s>[a, b] : a \<otimes> b \<rightarrow> b \<otimes> a\<guillemotright>"
   and sym_naturality: "\<lbrakk> arr f; arr g \<rbrakk> \<Longrightarrow> \<s>[cod f, cod g] \<cdot> (f \<otimes> g) = (g \<otimes> f) \<cdot> \<s>[dom f, dom g]"
   and sym_inverse: "\<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> inverse_arrows \<s>[a, b] \<s>[b, a]"
   and unitor_coherence: "ide a \<Longrightarrow> \<l>[a] \<cdot> \<s>[a, \<I>] = \<r>[a]"
   and assoc_coherence: "\<lbrakk> ide a; ide b; ide c \<rbrakk> \<Longrightarrow>
                           \<a>[b, c, a] \<cdot> \<s>[a, b \<otimes> c] \<cdot> \<a>[a, b, c]
                              = (b \<otimes> \<s>[a, c]) \<cdot> \<a>[b, a, c] \<cdot> (\<s>[a, b] \<otimes> c)"
   begin
 
     lemma sym_simps [simp]:
     assumes "ide a" and "ide b"
     shows "arr \<s>[a, b]"
     and "dom \<s>[a, b] = a \<otimes> b"
     and "cod \<s>[a, b] = b \<otimes> a"
       using assms sym_in_hom by auto
 
     interpretation monoidal_category C T \<alpha> \<iota>
       using induces_monoidal_category by simp
     interpretation CC: product_category C C ..
 
     interpretation S: symmetry_functor C C ..
     interpretation ToS: composite_functor CC.comp CC.comp C S.map T ..
 
     definition \<sigma> :: "'a * 'a \<Rightarrow> 'a"
     where "\<sigma> f \<equiv> if CC.arr f then \<s>[cod (fst f), cod (snd f)] \<cdot> (fst f \<otimes> snd f) else null"
 
     interpretation \<sigma>: natural_isomorphism CC.comp C T ToS.map \<sigma>
     proof -
       interpret \<sigma>: transformation_by_components CC.comp C T ToS.map "\<lambda>a. \<s>[fst a, snd a]"
         apply unfold_locales
         using sym_in_hom sym_naturality by auto
       interpret \<sigma>: natural_isomorphism CC.comp C T ToS.map \<sigma>.map
         apply unfold_locales
         using sym_inverse \<sigma>.map_simp_ide by auto
       have "\<sigma> = \<sigma>.map"
         using \<sigma>_def \<sigma>.map_def sym_naturality by fastforce
       thus "natural_isomorphism CC.comp C T ToS.map \<sigma>"
         using \<sigma>.natural_isomorphism_axioms by presburger
     qed
 
     interpretation symmetric_monoidal_category C T \<alpha> \<iota> \<sigma>
     proof
       show "\<And>a b. \<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> inverse_arrows (\<sigma> (a, b)) (\<sigma> (b, a))"
       proof -
         fix a b
         assume a: "ide a" and b: "ide b"
         show "inverse_arrows (\<sigma> (a, b)) (\<sigma> (b, a))"
           using a b sym_inverse comp_arr_dom \<sigma>_def by auto
       qed
       (*
        * TODO: Here just using "lunit" refers to the locale parameter, rather than
        * the constant introduced by the interpretation above of monoidal_category.
        * This is slightly mysterious.
        *)
       show "\<And>a. ide a \<Longrightarrow> local.lunit a \<cdot> \<sigma> (a, local.unity) = local.runit a"
       proof -
         fix a
         assume a: "ide a"
         show "local.lunit a \<cdot> \<sigma> (a, local.unity) = local.runit a"
           using a lunit_agreement \<I>_agreement sym_in_hom comp_arr_dom [of "\<s>[a, \<I>]"]
                 unitor_coherence runit_agreement \<sigma>_def
           by simp
       qed
       show "\<And>a b c. \<lbrakk> ide a; ide b; ide c \<rbrakk> \<Longrightarrow>
                       local.assoc b c a \<cdot> \<sigma> (a, local.tensor b c) \<cdot> local.assoc a b c
                         = local.tensor b (\<sigma> (a, c)) \<cdot> local.assoc b a c \<cdot> local.tensor (\<sigma> (a, b)) c"
       proof -
         fix a b c
         assume a: "ide a" and b: "ide b" and c: "ide c"
         show "local.assoc b c a \<cdot> \<sigma> (a, local.tensor b c) \<cdot> local.assoc a b c
                 = local.tensor b (\<sigma> (a, c)) \<cdot> local.assoc b a c \<cdot> local.tensor (\<sigma> (a, b)) c"
           using a b c sym_in_hom tensor_preserves_ide \<sigma>_def assoc_coherence
                 comp_arr_dom comp_cod_arr
           by simp
       qed
     qed
 
     lemma induces_symmetric_monoidal_category:
     shows "symmetric_monoidal_category C T \<alpha> \<iota> \<sigma>"
       ..
 
   end
 
   context symmetric_monoidal_category
   begin
 
     interpretation elementary_monoidal_category C tensor unity lunit runit assoc
       using induces_elementary_monoidal_category by auto
 
     lemma induces_elementary_symmetric_monoidal_category:
     shows "elementary_symmetric_monoidal_category
              C tensor unity lunit runit assoc (\<lambda>a b. \<sigma> (a, b))"
       using \<sigma>.naturality unitor_coherence assoc_coherence sym_inverse
       by unfold_locales auto
 
   end
 
   (* TODO: This definition of "diagonal_functor" conflicts with the one in Category3.Limit. *)
   locale diagonal_functor =
     C: category C +
     CC: product_category C C
   for C :: "'a comp"
   begin
 
     abbreviation map
     where "map f \<equiv> if C.arr f then (f, f) else CC.null"
 
     lemma is_functor:
     shows "functor C CC.comp map"
       using map_def by unfold_locales auto
 
     sublocale "functor" C CC.comp map
       using is_functor by simp
 
   end
 
   locale cartesian_monoidal_category =
     monoidal_category C T \<alpha> \<iota> +
     \<Omega>: constant_functor C C \<I> +
     \<Delta>: diagonal_functor C +
     \<tau>: natural_transformation C C map \<Omega>.map \<tau> +
     \<delta>: natural_transformation C C map \<open>T o \<Delta>.map\<close> \<delta>
   for C :: "'a comp"                            (infixr "\<cdot>" 55)
   and T :: "'a * 'a \<Rightarrow> 'a"
   and \<alpha> :: "'a * 'a * 'a \<Rightarrow> 'a"
   and \<iota> :: 'a
   and \<delta> :: "'a \<Rightarrow> 'a"                           ("\<d>[_]")
   and \<tau> :: "'a \<Rightarrow> 'a"                           ("\<t>[_]") +
   assumes trm_unity: "\<t>[\<I>] = \<I>"
   and pr0_dup: "ide a \<Longrightarrow> \<r>[a] \<cdot> (a \<otimes> \<t>[a]) \<cdot> \<delta> a = a"
   and pr1_dup: "ide a \<Longrightarrow> \<l>[a] \<cdot> (\<t>[a] \<otimes> a) \<cdot> \<delta> a = a"
   and tuple_pr: "\<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> (\<r>[a] \<cdot> (a \<otimes> \<t>[b]) \<otimes> \<l>[b] \<cdot> (\<t>[a] \<otimes> b)) \<cdot> \<d>[a \<otimes> b] = a \<otimes> b"
 
   locale elementary_cartesian_monoidal_category =
     elementary_monoidal_category C tensor unity lunit runit assoc
   for C :: "'a comp"                   (infixr "\<cdot>" 55)
   and tensor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"       (infixr "\<otimes>" 53)
   and unity :: 'a                      ("\<I>")
   and lunit :: "'a \<Rightarrow> 'a"              ("\<l>[_]")
   and runit :: "'a \<Rightarrow> 'a"              ("\<r>[_]")
   and assoc :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"  ("\<a>[_, _, _]")
   and trm :: "'a \<Rightarrow> 'a"                ("\<t>[_]")
   and dup :: "'a \<Rightarrow> 'a"                ("\<d>[_]") +
   assumes trm_in_hom [intro]: "ide a \<Longrightarrow> \<guillemotleft>\<t>[a] : a \<rightarrow> \<I>\<guillemotright>"
   and trm_unity: "\<t>[\<I>] = \<I>"
   and trm_naturality: "arr f \<Longrightarrow> \<t>[cod f] \<cdot> f = \<t>[dom f]"
   and dup_in_hom [intro]: "ide a \<Longrightarrow> \<guillemotleft>\<d>[a] : a \<rightarrow> a \<otimes> a\<guillemotright>"
   and dup_naturality: "arr f \<Longrightarrow> \<d>[cod f] \<cdot> f = (f \<otimes> f) \<cdot> \<d>[dom f]"
   and prj0_dup: "ide a \<Longrightarrow> \<r>[a] \<cdot> (a \<otimes> \<t>[a]) \<cdot> \<d>[a] = a"
   and prj1_dup: "ide a \<Longrightarrow> \<l>[a] \<cdot> (\<t>[a] \<otimes> a) \<cdot> \<d>[a] = a"
   and tuple_prj: "\<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow> (\<r>[a] \<cdot> (a \<otimes> \<t>[b]) \<otimes> \<l>[b] \<cdot> (\<t>[a] \<otimes> b)) \<cdot> \<d>[a \<otimes> b] = a \<otimes> b"
 
   context cartesian_monoidal_category
   begin
 
     lemma terminal_unity:
     shows "terminal \<I>"
     proof
       show "ide \<I>" by simp
       show "\<And>a. ide a \<Longrightarrow> \<exists>!f. \<guillemotleft>f : a \<rightarrow> \<I>\<guillemotright>"
       proof
         fix a
         assume a: "ide a"
         show "\<guillemotleft>\<tau> a : a \<rightarrow> \<I>\<guillemotright>" using a by auto
         show "\<And>f. \<guillemotleft>f : a \<rightarrow> \<I>\<guillemotright> \<Longrightarrow> f = \<tau> a"
           using trm_unity \<tau>.naturality comp_cod_arr by fastforce
       qed
     qed
 
     lemma trm_is_terminal_arr:
     assumes "ide a"
     shows "terminal_arr \<t>[a]"
       using assms terminal_unity by simp
 
     interpretation elementary_monoidal_category C tensor unity lunit runit assoc
       using induces_elementary_monoidal_category by simp
 
     interpretation elementary_cartesian_monoidal_category C tensor unity lunit runit assoc \<tau> \<delta>
     proof
       show "\<And>a. ide a \<Longrightarrow> \<guillemotleft>\<t>[a] : a \<rightarrow> \<I>\<guillemotright>"
         using \<iota>_in_hom by force
       show "\<t>[\<I>] = \<I>"
         using \<tau>.preserves_hom \<iota>_in_hom ide_unity trm_is_terminal_arr terminal_unity
         by (intro terminal_arr_unique) auto
       show "\<And>f. arr f \<Longrightarrow> \<t>[cod f] \<cdot> f = \<t>[dom f]"
         using \<tau>.naturality comp_cod_arr by simp
       show "\<And>a. ide a \<Longrightarrow> \<guillemotleft>\<d>[a] : a \<rightarrow> a \<otimes> a\<guillemotright>"
         by auto
       show "\<And>f. arr f \<Longrightarrow> \<d>[cod f] \<cdot> f = (f \<otimes> f) \<cdot> \<d>[dom f]"
         using \<delta>.naturality by simp
       show "\<And>a. ide a \<Longrightarrow> \<l>[a] \<cdot> (\<t>[a] \<otimes> a) \<cdot> \<d>[a] = a"
         using pr1_dup lunit_in_hom by simp
       show "\<And>a. ide a \<Longrightarrow> \<r>[a] \<cdot> (a \<otimes> \<t>[a]) \<cdot> \<d>[a] = a"
         using pr0_dup runit_in_hom by simp
       show "\<And>a0 a1. \<lbrakk> ide a0; ide a1 \<rbrakk> \<Longrightarrow>
                      (\<r>[a0] \<cdot> (a0 \<otimes> \<t>[a1]) \<otimes> \<l>[a1] \<cdot> (\<t>[a0] \<otimes> a1)) \<cdot> \<d>[a0 \<otimes> a1]
                        = a0 \<otimes> a1"
         using tuple_pr by simp
     qed
 
     lemma induces_elementary_cartesian_monoidal_category:
     shows "elementary_cartesian_monoidal_category C tensor \<I> lunit runit assoc \<tau> \<delta>"
       ..
 
   end
 
   context elementary_cartesian_monoidal_category
   begin
 
     lemma trm_simps [simp]:
     assumes "ide a"
     shows "arr \<t>[a]" and "dom \<t>[a] = a" and "cod \<t>[a] = \<I>"
       using assms trm_in_hom by auto
 
     lemma dup_simps [simp]:
     assumes "ide a"
     shows "arr \<d>[a]" and "dom \<d>[a] = a" and "cod \<d>[a] = a \<otimes> a"
       using assms dup_in_hom by auto
 
     definition \<tau> :: "'a \<Rightarrow> 'a"
     where "\<tau> f \<equiv> if arr f then \<t>[dom f] else null"
 
     definition \<delta> :: "'a \<Rightarrow> 'a"
     where "\<delta> f \<equiv> if arr f then \<d>[cod f] \<cdot> f else null"
 
     interpretation CC: product_category C C ..
 
     interpretation MC: monoidal_category C T \<alpha> \<iota>
       using induces_monoidal_category by auto
 
     interpretation I: constant_functor C C MC.unity
       by unfold_locales auto
     interpretation \<Delta>: diagonal_functor C ..
     interpretation D: composite_functor C CC.comp C \<Delta>.map T ..
     interpretation \<tau>: natural_transformation C C map I.map \<tau>
       using trm_naturality I.map_def \<tau>_def \<I>_agreement comp_cod_arr
       by unfold_locales auto
     interpretation \<delta>: natural_transformation C C map D.map \<delta>
       using dup_naturality \<delta>_def comp_arr_dom
       by unfold_locales auto
 
     interpretation MC: cartesian_monoidal_category C T \<alpha> \<iota> \<delta> \<tau>
     proof
       show "\<tau> MC.unity = MC.unity"
         using \<I>_agreement trm_unity \<tau>_def by simp
       show "\<And>a. ide a \<Longrightarrow> MC.runit a \<cdot> MC.tensor a (\<tau> a) \<cdot> \<delta> a = a"
         using runit_agreement \<tau>_def \<delta>_def prj0_dup comp_arr_dom by auto
       show "\<And>a. ide a \<Longrightarrow> MC.lunit a \<cdot> MC.tensor (\<tau> a) a \<cdot> \<delta> a = a"
         using lunit_agreement \<tau>_def \<delta>_def prj1_dup comp_arr_dom by auto
       show "\<And>a b. \<lbrakk> ide a; ide b \<rbrakk> \<Longrightarrow>
                   MC.tensor (MC.runit a \<cdot> MC.tensor a (\<tau> b)) (MC.lunit b \<cdot> MC.tensor (\<tau> a) b) \<cdot>
                   \<delta> (MC.tensor a b) = MC.tensor a b"
       proof -
         fix a b
         assume a: "ide a" and b: "ide b"
         have "seq \<r>[a] (a \<otimes> \<t>[b])"
           by (metis a b arr_tensor cod_tensor ide_char in_homE runit_in_hom seqI trm_simps(1,3))
         moreover have "seq \<l>[b] (\<t>[a] \<otimes> b)"
           by (metis a b arr_tensor cod_tensor ide_char in_homE lunit_in_hom seqI trm_simps(1,3))
         ultimately show "MC.tensor (MC.runit a \<cdot> MC.tensor a (\<tau> b))
                                    (MC.lunit b \<cdot> MC.tensor (\<tau> a) b) \<cdot> \<delta> (MC.tensor a b) =
                          MC.tensor a b"
           using a b lunit_agreement runit_agreement unitor_coincidence \<tau>_def \<delta>_def comp_arr_dom
                 tensor_preserves_ide tuple_prj T_def
           by auto
       qed
     qed
 
     lemma induces_cartesian_monoidal_category:
     shows "cartesian_monoidal_category C T \<alpha> \<iota> \<delta> \<tau>"
       ..
 
   end
 
   text \<open>
     A cartesian category extends to a a cartesian monoidal category by using the product
     structure to obtain the various canonical maps.
   \<close>
 
   context cartesian_category
   begin
 
     interpretation C: elementary_cartesian_category C pr0 pr1 \<one> trm
       using extends_to_elementary_cartesian_category by simp
     interpretation CC: product_category C C ..
     interpretation CCC: product_category C CC.comp ..
     interpretation T: binary_functor C C C Prod
       using binary_functor_Prod by simp
     interpretation T: binary_endofunctor C Prod ..
     interpretation ToTC: "functor" CCC.comp C T.ToTC
       using T.functor_ToTC by auto
     interpretation ToCT: "functor" CCC.comp C T.ToCT
       using T.functor_ToCT by auto
 
     interpretation \<alpha>: transformation_by_components CCC.comp C T.ToTC T.ToCT
                         \<open>\<lambda>abc. assoc (fst abc) (fst (snd abc)) (snd (snd abc))\<close>
     proof
       show "\<And>abc. CCC.ide abc \<Longrightarrow>
                      \<guillemotleft>assoc (fst abc) (fst (snd abc)) (snd (snd abc)) : T.ToTC abc \<rightarrow> T.ToCT abc\<guillemotright>"
         using CCC.ide_char CC.ide_char CCC.arr_char CC.arr_char T.ToTC_def T.ToCT_def
         by auto
       show "\<And>f. CCC.arr f \<Longrightarrow>
                   assoc (fst (CCC.cod f)) (fst (snd (CCC.cod f))) (snd (snd (CCC.cod f)))
                           \<cdot> T.ToTC f =
                   T.ToCT f
                     \<cdot> assoc (fst (CCC.dom f)) (fst (snd (CCC.dom f))) (snd (snd (CCC.dom f)))"
         using CCC.arr_char CC.arr_char CCC.dom_char CCC.cod_char T.ToTC_def T.ToCT_def
               assoc_naturality
         by simp blast
     qed
 
     interpretation \<alpha>: natural_isomorphism CCC.comp C T.ToTC T.ToCT \<alpha>.map
     proof
       show "\<And>a. CCC.ide a \<Longrightarrow> iso (\<alpha>.map a)"
         using CCC.ide_char CC.ide_char \<alpha>.map_simp_ide inverse_arrows_assoc by auto
     qed
 
     interpretation L: "functor" C C \<open>\<lambda>f. Prod (cod \<iota>, f)\<close>
       using \<iota>_is_terminal_arr T.fixing_ide_gives_functor_1 by simp
     interpretation L: endofunctor C \<open>\<lambda>f. Prod (cod \<iota>, f)\<close> ..
     interpretation \<l>: transformation_by_components C C
                         \<open>\<lambda>f. Prod (cod \<iota>, f)\<close> map \<open>\<lambda>a. pr0 (cod \<iota>) a\<close>
       using \<iota>_is_terminal_arr
       by unfold_locales auto
     interpretation \<l>: natural_isomorphism C C \<open>\<lambda>f. Prod (cod \<iota>, f)\<close> map \<l>.map
       using \<l>.map_simp_ide inverse_arrows_lunit ide_some_terminal
       by unfold_locales auto
     interpretation L: equivalence_functor C C \<open>\<lambda>f. Prod (cod \<iota>, f)\<close>
       using \<l>.natural_isomorphism_axioms naturally_isomorphic_def
             L.isomorphic_to_identity_is_equivalence
       by blast
 
     interpretation R: "functor" C C \<open>\<lambda>f. Prod (f, cod \<iota>)\<close>
       using \<iota>_is_terminal_arr T.fixing_ide_gives_functor_2 by simp
     interpretation R: endofunctor C\<open>\<lambda>f. Prod (f, cod \<iota>)\<close> ..
     interpretation \<rho>: transformation_by_components C C
                         \<open>\<lambda>f. Prod (f, cod \<iota>)\<close> map \<open>\<lambda>a. pr1 a (cod \<iota>)\<close>
       using \<iota>_is_terminal_arr
       by unfold_locales auto
     interpretation \<rho>: natural_isomorphism C C \<open>\<lambda>f. Prod (f, cod \<iota>)\<close> map \<rho>.map
       using \<rho>.map_simp_ide inverse_arrows_runit ide_some_terminal
       by unfold_locales auto
     interpretation R: equivalence_functor C C \<open>\<lambda>f. Prod (f, cod \<iota>)\<close>
       using \<rho>.natural_isomorphism_axioms naturally_isomorphic_def
             R.isomorphic_to_identity_is_equivalence
       by blast
 
     interpretation C: monoidal_category C Prod \<alpha>.map \<iota>
       using ide_some_terminal \<iota>_is_iso pentagon comp_assoc
       by unfold_locales auto
 
     interpretation \<Omega>: constant_functor C C C.unity
       using C.ide_unity
       by unfold_locales auto
 
     interpretation \<tau>: natural_transformation C C map \<Omega>.map trm
       using C.unity_def \<Omega>.map_def ide_some_terminal trm_naturality comp_cod_arr trm_in_hom
       apply unfold_locales
       using trm_def
           apply auto[1]
          apply fastforce
         apply fastforce
        apply (metis in_homE trm_eqI trm_in_hom cod_pr0 dom_dom)
       by (metis trm_eqI trm_in_hom dom_dom map_simp)
 
     interpretation \<Delta>: "functor" C CC.comp Diag
       using functor_Diag by simp
     interpretation \<Pi>o\<Delta>: composite_functor C CC.comp C Diag Prod ..
 
     interpretation natural_transformation C C map \<open>Prod o Diag\<close> dup
       using dup_is_natural_transformation by simp
 
     lemma unity_agreement:
     shows "C.unity = \<one>"
       using C.unity_def ide_some_terminal by simp
 
     lemma assoc_agreement:
     assumes "ide a" and "ide b" and "ide c"
     shows "C.assoc a b c = assoc a b c"
       using assms assoc_def \<alpha>.map_simp_ide by simp
 
     lemma assoc'_agreement:
     assumes "ide a" and "ide b" and "ide c"
     shows "C.assoc' a b c = assoc' a b c"
       using assms inverse_arrows_assoc inverse_unique by auto
 
     lemma runit_char_eqn:
     assumes "ide a"
     shows "prod (runit a) \<one> = prod a \<iota> \<cdot> assoc a \<one> \<one>"
       using assms ide_one assoc_def comp_assoc prod_tuple comp_cod_arr
       by (intro pr_joint_monic [of a \<one> "prod (runit a) \<one>" "prod a \<iota> \<cdot> assoc a \<one> \<one>"]) auto
 
     lemma runit_agreement:
     assumes "ide a"
     shows "runit a = C.runit a"
       using assms unity_agreement assoc_agreement C.runit_char(2) runit_char_eqn
             ide_some_terminal
       by (intro C.runit_eqI) auto
 
     lemma lunit_char_eqn:
     assumes "ide a"
     shows "prod \<one> (lunit a) = prod \<iota> a \<cdot> assoc' \<one> \<one> a"
     proof (intro pr_joint_monic [of \<one> a "prod \<one> (lunit a)" "prod \<iota> a \<cdot> assoc' \<one> \<one> a"])
       show "seq (lunit a) (local.prod \<one> (lunit a))"
         using assms ide_one by simp
       show "lunit a \<cdot> prod \<one> (lunit a) = lunit a \<cdot> prod \<iota> a \<cdot> assoc' \<one> \<one> a"
         using assms ide_one assoc'_def comp_assoc prod_tuple comp_cod_arr by simp
       show "pr1 \<one> a \<cdot> prod \<one> (lunit a) = pr1 \<one> a \<cdot> prod \<iota> a \<cdot> assoc' \<one> \<one> a"
         using assms ide_one assoc'_def comp_cod_arr prod_tuple pr_naturality
         apply simp
         by (metis trm_eqI cod_pr0 cod_pr1 comp_in_homI' ide_prod pr_simps(1,3-6) pr1_in_hom')
     qed
 
     lemma lunit_agreement:
     assumes "ide a"
     shows "lunit a = C.lunit a"
       using assms unity_agreement assoc'_agreement C.lunit_char(2) lunit_char_eqn
             ide_some_terminal
       by (intro C.lunit_eqI) auto
 
     interpretation C: cartesian_monoidal_category C Prod \<alpha>.map \<iota> dup trm
     proof
       show "trm C.unity = C.unity"
         by (simp add: C.unity_def ide_some_terminal trm_one)
       show "\<And>a. ide a \<Longrightarrow> C.runit a \<cdot> C.tensor a \<t>[a] \<cdot> dup a = a"
         using comp_runit_term_dup runit_agreement by simp
       show "\<And>a. ide a \<Longrightarrow> C.lunit a \<cdot> C.tensor \<t>[a] a \<cdot> dup a = a"
         using comp_lunit_term_dup lunit_agreement by auto
       show "\<And>a b. \<lbrakk>ide a; ide b\<rbrakk>
                      \<Longrightarrow> C.tensor (C.runit a \<cdot> C.tensor a \<t>[b])
                                   (C.lunit b \<cdot> C.tensor \<t>[a] b) \<cdot> dup (C.tensor a b) =
                          C.tensor a b"
       proof -
         fix a b
         assume a: "ide a" and b: "ide b"
         have "C.tensor (C.runit a \<cdot> C.tensor a \<t>[b])
                        (C.lunit b \<cdot> C.tensor \<t>[a] b) \<cdot> dup (C.tensor a b) =
               prod (C.runit a \<cdot> prod a \<t>[b]) (C.lunit b \<cdot> prod \<t>[a] b) \<cdot> dup (prod a b)"
           using a b by simp
         also have "... = tuple ((C.runit a \<cdot> prod a \<t>[b]) \<cdot> prod a b)
                                ((C.lunit b \<cdot> prod \<t>[a] b) \<cdot> prod a b)"
           using a b ide_one trm_in_hom [of a] trm_in_hom [of b] unity_agreement prod_tuple
           by fastforce
         also have "... = tuple (C.runit a \<cdot> prod a \<t>[b] \<cdot> prod a b)
                                (C.lunit b \<cdot> prod \<t>[a] b \<cdot> prod a b)"
           using comp_assoc by simp
         also have "... = tuple (C.runit a \<cdot> prod a \<t>[b]) (C.lunit b \<cdot> prod \<t>[a] b)"
           using a b comp_arr_dom by simp
         also have "... = tuple (runit a \<cdot> prod a \<t>[b]) (lunit b \<cdot> prod \<t>[a] b)"
           using a b lunit_agreement runit_agreement by simp
         also have "... = tuple (pr1 a b) (pr0 a b)"
         proof -
           have "runit a \<cdot> prod a \<t>[b] = pr1 a b"
             using a b pr_naturality(2) trm_in_hom [of b]
             by (metis cod_pr1 comp_ide_arr ide_char in_homE pr_simps(4,6) seqI)
           moreover have "lunit b \<cdot> prod \<t>[a] b = pr0 a b"
             using a b pr_naturality(1) [of b b b "\<t>[a]" a \<one>] trm_in_hom [of a] comp_cod_arr
             by (metis cod_pr0 ide_char in_homE pr_simps(1))
           ultimately show ?thesis by simp
         qed
         also have "... = prod a b"
           using a b by simp
         finally show "C.tensor (C.runit a \<cdot> C.tensor a \<t>[b])
                                (C.lunit b \<cdot> C.tensor \<t>[a] b) \<cdot> dup (C.tensor a b) =
                       C.tensor a b"
           by auto
       qed
     qed
 
     lemma extends_to_cartesian_monoidal_category:
     shows "cartesian_monoidal_category C Prod \<alpha>.map \<iota> dup trm"
       ..
 
   end
 
   text \<open>
     In a \<open>cartesian_monoidal_category\<close>, the monoidal structure is given by a categorical product
     and terminal object, so that the underlying category is cartesian.
   \<close>
 
   context cartesian_monoidal_category
   begin
 
     definition pr0                        ("\<p>\<^sub>0[_, _]")
     where "\<p>\<^sub>0[a, b] \<equiv> if ide a \<and> ide b then \<r>[a] \<cdot> (a \<otimes> \<t>[b]) else null"
 
     definition pr1                        ("\<p>\<^sub>1[_, _]")
     where "\<p>\<^sub>1[a, b] \<equiv> if ide a \<and> ide b then \<l>[b] \<cdot> (\<t>[a] \<otimes> b) else null"
 
     lemma pr_in_hom [intro]:
     assumes "ide a0" and "ide a1"
     shows "\<guillemotleft>\<p>\<^sub>0[a0, a1] : a0 \<otimes> a1 \<rightarrow> a0\<guillemotright>"
     and "\<guillemotleft>\<p>\<^sub>1[a0, a1] : a0 \<otimes> a1 \<rightarrow> a1\<guillemotright>"
     proof -
       show "\<guillemotleft>\<p>\<^sub>0[a0, a1] : a0 \<otimes> a1 \<rightarrow> a0\<guillemotright>"
         unfolding pr0_def
         using assms runit_in_hom by fastforce
       show "\<guillemotleft>\<p>\<^sub>1[a0, a1] : a0 \<otimes> a1 \<rightarrow> a1\<guillemotright>"
         unfolding pr1_def
         using assms lunit_in_hom by fastforce
     qed
 
     lemma pr_simps [simp]:
     assumes "ide a0" and "ide a1"
     shows "arr \<p>\<^sub>0[a0, a1]" and "dom \<p>\<^sub>0[a0, a1] = a0 \<otimes> a1" and "cod \<p>\<^sub>0[a0, a1] = a0"
     and "arr \<p>\<^sub>1[a0, a1]" and "dom \<p>\<^sub>1[a0, a1] = a0 \<otimes> a1" and "cod \<p>\<^sub>1[a0, a1] = a1"
       using assms pr_in_hom(1-2) by blast+
 
     interpretation P: composite_functor CC.comp C CC.comp T \<Delta>.map ..
 
     interpretation ECC: elementary_cartesian_category C pr1 pr0 \<I> \<tau>
     proof
       show "\<And>a b. \<lbrakk>ide a; ide b\<rbrakk> \<Longrightarrow> span \<p>\<^sub>0[a, b] \<p>\<^sub>1[a, b]"
         by simp
       show "\<And>a b. \<lbrakk>ide a; ide b\<rbrakk> \<Longrightarrow> cod \<p>\<^sub>1[a, b] = b"
         by simp
       show "\<And>a b. \<lbrakk>ide a; ide b\<rbrakk> \<Longrightarrow> cod \<p>\<^sub>0[a, b] = a"
         by simp
       show "ide \<I>"
         by simp
       show "\<And>a. ide a \<Longrightarrow> \<guillemotleft>\<tau> a : a \<rightarrow> \<I>\<guillemotright>"
         by auto
       show "\<And>a f. \<lbrakk>ide a; \<guillemotleft>f : a \<rightarrow> \<I>\<guillemotright>\<rbrakk> \<Longrightarrow> f = \<t>[a]"
         using \<open>\<And>a. ide a \<Longrightarrow> \<guillemotleft>\<t>[a] : a \<rightarrow> \<I>\<guillemotright>\<close> terminalE terminal_unity by blast
       show "\<And>a b. \<not> (ide a \<and> ide b) \<Longrightarrow> \<p>\<^sub>1[a, b] = null"
         using pr1_def by auto
       show "\<And>a b. \<not> (ide a \<and> ide b) \<Longrightarrow> \<p>\<^sub>0[a, b] = null"
         using pr0_def by auto
       show "\<And>f g. span f g \<Longrightarrow> \<exists>!l. \<p>\<^sub>0[cod f, cod g] \<cdot> l = f \<and> \<p>\<^sub>1[cod f, cod g] \<cdot> l = g"
       proof -
         fix f g
         assume fg: "span f g"
         let ?l = "(f \<otimes> g) \<cdot> \<d>[dom f]"
         have "\<p>\<^sub>0[cod f, cod g] \<cdot> ?l = f"
         proof -
           have "\<p>\<^sub>0[cod f, cod g] \<cdot> ?l = (\<r>[cod f] \<cdot> (cod f \<otimes> \<t>[cod g])) \<cdot> (f \<otimes> g) \<cdot> \<d>[dom f]"
             using fg pr0_def by simp
           also have "... = \<r>[cod f] \<cdot> ((cod f \<otimes> \<t>[cod g]) \<cdot> (f \<otimes> g)) \<cdot> \<d>[dom f]"
             using comp_assoc by simp
           also have "... = \<r>[cod f] \<cdot> (f \<otimes> \<t>[dom f]) \<cdot> \<d>[dom f]"
             using fg interchange comp_cod_arr \<tau>.naturality by simp
           also have "... = \<r>[cod f] \<cdot> ((cod f \<otimes> \<t>[cod f]) \<cdot> (f \<otimes> f)) \<cdot> \<d>[dom f]"
             using fg interchange comp_cod_arr \<tau>.naturality by simp
           also have "... = \<r>[cod f] \<cdot> (cod f \<otimes> \<t>[cod f]) \<cdot> (f \<otimes> f) \<cdot> \<d>[dom f]"
             using comp_assoc by simp
           also have "... = (\<r>[cod f] \<cdot> (cod f \<otimes> \<t>[cod f]) \<cdot> \<d>[cod f]) \<cdot> f"
             using fg \<delta>.naturality comp_assoc by simp
           also have "... = f"
             using fg pr0_dup comp_cod_arr by simp
           finally show ?thesis by blast
         qed
         moreover have "\<p>\<^sub>1[cod f, cod g] \<cdot> ?l = g"
         proof -
           have "\<p>\<^sub>1[cod f, cod g] \<cdot> ?l = \<l>[cod g] \<cdot> ((\<t>[cod f] \<otimes> cod g) \<cdot> (f \<otimes> g)) \<cdot> \<d>[dom g]"
             using fg pr1_def comp_assoc by simp
           also have "... = \<l>[cod g] \<cdot> (\<t>[dom f] \<otimes> g) \<cdot> \<d>[dom g]"
             using fg interchange comp_cod_arr \<tau>.naturality
             by simp
           also have "... = \<l>[cod g] \<cdot> ((\<t>[cod g] \<otimes> cod g) \<cdot> (g \<otimes> g)) \<cdot> \<d>[dom g]"
             using fg interchange comp_cod_arr \<tau>.naturality by simp
           also have "... = \<l>[cod g] \<cdot> (\<t>[cod g] \<otimes> cod g) \<cdot> (g \<otimes> g) \<cdot> \<d>[dom g]"
             using comp_assoc by simp
           also have "... = (\<l>[cod g] \<cdot> (\<t>[cod g] \<otimes> cod g) \<cdot> \<d>[cod g]) \<cdot> g"
             using fg \<delta>.naturality comp_assoc by simp
           also have "... = g"
             using fg pr1_dup comp_cod_arr by simp
           finally show ?thesis by blast
         qed
         moreover have "\<And>l. \<lbrakk> \<p>\<^sub>0[cod f, cod g] \<cdot> l = f; \<p>\<^sub>1[cod f, cod g] \<cdot> l = g \<rbrakk> \<Longrightarrow> l = ?l"
         proof -
           fix l
           assume f: "\<p>\<^sub>0[cod f, cod g] \<cdot> l = f" and g: "\<p>\<^sub>1[cod f, cod g] \<cdot> l = g"
           have l: "\<guillemotleft>l : dom f \<rightarrow> cod f \<otimes> cod g\<guillemotright>"
             using f g fg
             by (metis arr_iff_in_hom dom_comp ide_cod pr_simps(5) seqE)
           have "?l = (\<p>\<^sub>0[cod f, cod g] \<cdot> l \<otimes> \<p>\<^sub>1[cod f, cod g] \<cdot> l) \<cdot> \<d>[dom f]"
             using f g by simp
           also have "... = ((\<p>\<^sub>0[cod f, cod g] \<otimes> \<p>\<^sub>1[cod f, cod g]) \<cdot> (l \<otimes> l)) \<cdot> \<d>[dom f]"
             using fg f g interchange by simp
           also have "... = (\<p>\<^sub>0[cod f, cod g] \<otimes> \<p>\<^sub>1[cod f, cod g]) \<cdot> (l \<otimes> l) \<cdot> \<d>[dom f]"
             using comp_assoc by simp
           also have "... = ((\<p>\<^sub>0[cod f, cod g] \<otimes> \<p>\<^sub>1[cod f, cod g]) \<cdot> \<d>[cod f \<otimes> cod g]) \<cdot> l"
             using l \<delta>.naturality [of l] comp_assoc by auto
           also have "... = (cod f \<otimes> cod g) \<cdot> l"
             using f g fg tuple_pr pr0_def pr1_def by auto
           also have "... = l"
             using l comp_cod_arr by auto
           finally show "l = ?l" by simp
         qed
         ultimately show "\<exists>!l. \<p>\<^sub>0[cod f, cod g] \<cdot> l = f \<and> \<p>\<^sub>1[cod f, cod g] \<cdot> l = g"
           by auto
       qed
     qed
 
     lemma extends_to_elementary_cartesian_category:
     shows "elementary_cartesian_category C pr1 pr0 \<I> \<tau>"
       ..
 
     lemma is_cartesian_category:
     shows "cartesian_category C"
       using ECC.is_cartesian_category by simp
 
   end
 
   (*
    * TODO: I would like to have coherence theorems for symmetric monoidal and cartesian
    * monoidal categories here, but I haven't yet figured out a suitably economical way
    * to extend the existing result.
    *)
 
 end