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	(* Title: Pseudofunctor
	Author: Eugene W. Stark <stark@cs.stonybrook.edu>, 2019
	Maintainer: Eugene W. Stark <stark@cs.stonybrook.edu>
	*)

	section "Pseudofunctors"

	theory Pseudofunctor
	imports MonoidalCategory.MonoidalFunctor Bicategory Subbicategory InternalEquivalence Coherence
	begin

	text \<open>
	The traditional definition of a pseudofunctor \<open>F : C \<rightarrow> D\<close> between bicategories \<open>C\<close> and \<open>D\<close>
	is in terms of two maps: an ``object map'' \<open>F\<^sub>o\<close> that takes objects of \<open>C\<close> to objects of \<open>D\<close>
	and an ``arrow map'' \<open>F\<^sub>a\<close> that assigns to each pair of objects \<open>a\<close> and \<open>b\<close> of \<open>C\<close>
	a functor \<open>F\<^sub>a a b\<close> from the hom-category \<open>hom\<^sub>C a b\<close> to the hom-category \<open>hom\<^sub>D (F\<^sub>o a) (F\<^sub>o b)\<close>.
	In addition, there is assigned to each object \<open>a\<close> of \<open>C\<close> an invertible 2-cell
	\<open>\<guillemotleft>\<Psi> a : F\<^sub>o a \<Rightarrow>\<^sub>D (F\<^sub>a a a) a\<guillemotright>\<close>, and to each pair \<open>(f, g)\<close> of composable 1-cells of C there
	is assigned an invertible 2-cell \<open>\<guillemotleft>\<Phi> (f, g) : F g \<star> F f \<Rightarrow> F (g \<star> f)\<guillemotright>\<close>, all subject to
	naturality and coherence conditions.

	In keeping with the ``object-free'' style in which we have been working, we do not wish
	to adopt a definition of pseudofunctor that distinguishes between objects and other
	arrows. Instead, we would like to understand a pseudofunctor as an ordinary functor between
	(vertical) categories that weakly preserves horizontal composition in a suitable sense.
	So, we take as a starting point that a pseudofunctor \<open>F : C \<rightarrow> D\<close> is a functor from
	\<open>C\<close> to \<open>D\<close>, when these are regarded as ordinary categories with respect to vertical
	composition. Next, \<open>F\<close> should preserve source and target, but only ``weakly''
	(up to isomorphism, rather than ``on the nose'').
	Weak preservation of horizontal composition is expressed by specifying, for each horizontally
	composable pair of vertical identities \<open>(f, g)\<close> of \<open>C\<close>, a ``compositor''
	\<open>\<guillemotleft>\<Phi> (f, g) : F g \<star> F f \<Rightarrow> F (g \<star> f)\<guillemotright>\<close> in \<open>D\<close>, such that the \<open>\<Phi> (f, g)\<close> are the components
	of a natural isomorphism.
	Associators must also be weakly preserved by F; this is expressed by a coherence
	condition that relates an associator \<open>\<a>\<^sub>C[f, g, h]\<close> in \<open>C\<close>, its image \<open>F \<a>\<^sub>C[f, g, h]\<close>,
	the associator \<open>\<a>\<^sub>D[F f, F g, F h]\<close> in \<open>D\<close> and compositors involving \<open>f\<close>, \<open>g\<close>, and \<open>h\<close>.
	As regards the weak preservation of unitors, just as for monoidal functors,
	which are in fact pseudofunctors between one-object bicategories, it is only necessary
	to assume that \<open>F \<i>\<^sub>C[a]\<close> and \<open>\<i>\<^sub>D[F a]\<close> are isomorphic in \<open>D\<close> for each object \<open>a\<close> of \<open>C\<close>,
	for there is then a canonical way to obtain, for each \<open>a\<close>, an isomorphism
	\<open>\<guillemotleft>\<Psi> a : src (F a) \<rightarrow> F a\<guillemotright>\<close> that satisfies the usual coherence conditions relating the
	unitors and the associators. Note that the map \<open>a \<mapsto> src (F a)\<close> amounts to the traditional
	``object map'' \<open>F\<^sub>o\<close>, so that this becomes a derived notion, rather than a primitive one.
	\<close>

	subsection "Weak Arrows of Homs"

	text \<open>
	We begin with a locale that defines a functor between ``horizontal homs'' that preserves
	source and target up to isomorphism.
	\<close>

	locale weak_arrow_of_homs =
	C: horizontal_homs C src\<^sub>C trg\<^sub>C +
	D: horizontal_homs D src\<^sub>D trg\<^sub>D +
	"functor" C D F
	for C :: "'c comp" (infixr "\<cdot>\<^sub>C" 55)
	and src\<^sub>C :: "'c \<Rightarrow> 'c"
	and trg\<^sub>C :: "'c \<Rightarrow> 'c"
	and D :: "'d comp" (infixr "\<cdot>\<^sub>D" 55)
	and src\<^sub>D :: "'d \<Rightarrow> 'd"
	and trg\<^sub>D :: "'d \<Rightarrow> 'd"
	and F :: "'c \<Rightarrow> 'd" +
	assumes weakly_preserves_src: "\<And>\<mu>. C.arr \<mu> \<Longrightarrow> D.isomorphic (F (src\<^sub>C \<mu>)) (src\<^sub>D (F \<mu>))"
	and weakly_preserves_trg: "\<And>\<mu>. C.arr \<mu> \<Longrightarrow> D.isomorphic (F (trg\<^sub>C \<mu>)) (trg\<^sub>D (F \<mu>))"
	begin

	lemma isomorphic_src:
	assumes "C.obj a"
	shows "D.isomorphic (src\<^sub>D (F a)) (F a)"
	using assms weakly_preserves_src [of a] D.isomorphic_symmetric by auto

	lemma isomorphic_trg:
	assumes "C.obj a"
	shows "D.isomorphic (trg\<^sub>D (F a)) (F a)"
	using assms weakly_preserves_trg [of a] D.isomorphic_symmetric by auto

	abbreviation (input) hseq\<^sub>C
	where "hseq\<^sub>C \<mu> \<nu> \<equiv> C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<mu> = trg\<^sub>C \<nu>"

	abbreviation (input) hseq\<^sub>D
	where "hseq\<^sub>D \<mu> \<nu> \<equiv> D.arr \<mu> \<and> D.arr \<nu> \<and> src\<^sub>D \<mu> = trg\<^sub>D \<nu>"

	lemma preserves_hseq:
	assumes "hseq\<^sub>C \<mu> \<nu>"
	shows "hseq\<^sub>D (F \<mu>) (F \<nu>)"
	by (metis D.isomorphic_def D.src_src D.src_trg D.vconn_implies_hpar(3)
	assms preserves_reflects_arr weakly_preserves_src weakly_preserves_trg)

	text \<open>
	Though \<open>F\<close> does not preserve objects ``on the nose'', we can recover from it the
	usual ``object map'', which does.
	It is slightly confusing at first to get used to the idea that applying the
	object map of a weak arrow of homs to an object does not give the same thing
	as applying the underlying functor, but rather only something isomorphic to it.

	The following defines the object map associated with \<open>F\<close>.
	\<close>

	definition map\<^sub>0
	where "map\<^sub>0 a \<equiv> src\<^sub>D (F a)"

	lemma map\<^sub>0_simps [simp]:
	assumes "C.obj a"
	shows "D.obj (map\<^sub>0 a)"
	and "src\<^sub>D (map\<^sub>0 a) = map\<^sub>0 a" and "trg\<^sub>D (map\<^sub>0 a) = map\<^sub>0 a"
	and "D.dom (map\<^sub>0 a) = map\<^sub>0 a" and "D.cod (map\<^sub>0 a) = map\<^sub>0 a"
	using assms map\<^sub>0_def by auto

	lemma preserves_src [simp]:
	assumes "C.arr \<mu>"
	shows "src\<^sub>D (F \<mu>) = map\<^sub>0 (src\<^sub>C \<mu>)"
	using assms
	by (metis C.src.preserves_arr C.src_src C.trg_src map\<^sub>0_def preserves_hseq)

	lemma preserves_trg [simp]:
	assumes "C.arr \<mu>"
	shows "trg\<^sub>D (F \<mu>) = map\<^sub>0 (trg\<^sub>C \<mu>)"
	using assms map\<^sub>0_def preserves_hseq C.src_trg C.trg.preserves_arr by presburger

	lemma preserves_hhom [intro]:
	assumes "C.arr \<mu>"
	shows "D.in_hhom (F \<mu>) (map\<^sub>0 (src\<^sub>C \<mu>)) (map\<^sub>0 (trg\<^sub>C \<mu>))"
	using assms by simp

	text \<open>
	We define here the lifting of \<open>F\<close> to a functor \<open>FF: CC \<rightarrow> DD\<close>.
	We need this to define the domains and codomains of the compositors.
	\<close>

	definition FF
	where "FF \<equiv> \<lambda>\<mu>\<nu>. if C.VV.arr \<mu>\<nu> then (F (fst \<mu>\<nu>), F (snd \<mu>\<nu>)) else D.VV.null"

	sublocale FF: "functor" C.VV.comp D.VV.comp FF
	proof -
	have 1: "\<And>\<mu>\<nu>. C.VV.arr \<mu>\<nu> \<Longrightarrow> D.VV.arr (FF \<mu>\<nu>)"
	unfolding FF_def using C.VV.arr_char D.VV.arr_char preserves_hseq by simp
	show "functor C.VV.comp D.VV.comp FF"
	proof
	fix \<mu>\<nu>
	show "\<not> C.VV.arr \<mu>\<nu> \<Longrightarrow> FF \<mu>\<nu> = D.VV.null"
	using FF_def by simp
	show "C.VV.arr \<mu>\<nu> \<Longrightarrow> D.VV.arr (FF \<mu>\<nu>)"
	using 1 by simp
	assume \<mu>\<nu>: "C.VV.arr \<mu>\<nu>"
	show "D.VV.dom (FF \<mu>\<nu>) = FF (C.VV.dom \<mu>\<nu>)"
	using \<mu>\<nu> 1 FF_def C.VV.arr_char D.VV.arr_char C.VV.dom_simp D.VV.dom_simp
	by simp
	show "D.VV.cod (FF \<mu>\<nu>) = FF (C.VV.cod \<mu>\<nu>)"
	using \<mu>\<nu> 1 FF_def C.VV.arr_char D.VV.arr_char C.VV.cod_simp D.VV.cod_simp
	by simp
	next
	fix \<mu>\<nu> \<tau>\<pi>
	assume 2: "C.VV.seq \<mu>\<nu> \<tau>\<pi>"
	show "FF (C.VV.comp \<mu>\<nu> \<tau>\<pi>) = D.VV.comp (FF \<mu>\<nu>) (FF \<tau>\<pi>)"
	proof -
	have "FF (C.VV.comp \<mu>\<nu> \<tau>\<pi>) = (F (fst \<mu>\<nu>) \<cdot>\<^sub>D F (fst \<tau>\<pi>), F (snd \<mu>\<nu>) \<cdot>\<^sub>D F (snd \<tau>\<pi>))"
	using 1 2 FF_def C.VV.comp_char C.VxV.comp_char C.VV.arr_char
	by (metis (no_types, lifting) C.VV.seq_char C.VxV.seqE fst_conv
	as_nat_trans.preserves_comp_2 snd_conv)
	also have "... = D.VV.comp (FF \<mu>\<nu>) (FF \<tau>\<pi>)"
	using 1 2 FF_def D.VV.comp_char D.VxV.comp_char C.VV.arr_char D.VV.arr_char
	C.VV.seq_char C.VxV.seqE preserves_seq
	by (simp, meson)
	finally show ?thesis by simp
	qed
	qed
	qed

	lemma functor_FF:
	shows "functor C.VV.comp D.VV.comp FF"
	..

	end

	subsection "Definition of Pseudofunctors"

	text \<open>
	I don't much like the term "pseudofunctor", which is suggestive of something that
	is ``not really'' a functor. In the development here we can see that a pseudofunctor
	is really a \emph{bona fide} functor with respect to vertical composition,
	which happens to have in addition a weak preservation property with respect to
	horizontal composition.
	This weak preservation of horizontal composition is captured by extra structure,
	the ``compositors'', which are the components of a natural transformation.
	So ``pseudofunctor'' is really a misnomer; it's an actual functor that has been equipped
	with additional structure relating to horizontal composition. I would use the term
	``bifunctor'' for such a thing, but it seems to not be generally accepted and also tends
	to conflict with the usage of that term to refer to an ordinary functor of two
	arguments; which I have called a ``binary functor''. Sadly, there seem to be no other
	plausible choices of terminology, other than simply ``functor''
	(recommended on n-Lab @{url \<open>https://ncatlab.org/nlab/show/pseudofunctor\<close>}),
	but that is not workable here because we need a name that does not clash with that
	used for an ordinary functor between categories.
	\<close>

	locale 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 +
	weak_arrow_of_homs V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F +
	FoH\<^sub>C: composite_functor C.VV.comp V\<^sub>C V\<^sub>D \<open>\<lambda>\<mu>\<nu>. H\<^sub>C (fst \<mu>\<nu>) (snd \<mu>\<nu>)\<close> F +
	H\<^sub>DoFF: composite_functor C.VV.comp D.VV.comp V\<^sub>D
	FF \<open>\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>)\<close> +
	\<Phi>: natural_isomorphism C.VV.comp V\<^sub>D H\<^sub>DoFF.map FoH\<^sub>C.map \<Phi>
	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> :: "'c * 'c \<Rightarrow> 'd" +
	assumes assoc_coherence:
	"\<lbrakk> C.ide f; C.ide g; C.ide h; src\<^sub>C f = trg\<^sub>C g; src\<^sub>C g = trg\<^sub>C h \<rbrakk> \<Longrightarrow>
	F \<a>\<^sub>C[f, g, h] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h) =
	\<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h]"
	begin

	no_notation C.in_hom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>C _\<guillemotright>")
	no_notation D.in_hom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>D _\<guillemotright>")
	notation C.in_hhom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>C _\<guillemotright>")
	notation C.in_hom ("\<guillemotleft>_ : _ \<Rightarrow>\<^sub>C _\<guillemotright>")
	notation D.in_hhom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>D _\<guillemotright>")
	notation D.in_hom ("\<guillemotleft>_ : _ \<Rightarrow>\<^sub>D _\<guillemotright>")

	notation C.lunit ("\<l>\<^sub>C[_]")
	notation C.runit ("\<r>\<^sub>C[_]")
	notation C.lunit' ("\<l>\<^sub>C\<^sup>-\<^sup>1[_]")
	notation C.runit' ("\<r>\<^sub>C\<^sup>-\<^sup>1[_]")
	notation C.\<a>' ("\<a>\<^sub>C\<^sup>-\<^sup>1[_, _, _]")
	notation D.lunit ("\<l>\<^sub>D[_]")
	notation D.runit ("\<r>\<^sub>D[_]")
	notation D.lunit' ("\<l>\<^sub>D\<^sup>-\<^sup>1[_]")
	notation D.runit' ("\<r>\<^sub>D\<^sup>-\<^sup>1[_]")
	notation D.\<a>' ("\<a>\<^sub>D\<^sup>-\<^sup>1[_, _, _]")

	lemma weakly_preserves_objects:
	assumes "C.obj a"
	shows "D.isomorphic (map\<^sub>0 a) (F a)"
	using assms weakly_preserves_src [of a] D.isomorphic_symmetric by auto

	lemma cmp_in_hom [intro]:
	assumes "C.ide a" and "C.ide b" and "src\<^sub>C a = trg\<^sub>C b"
	shows "\<guillemotleft>\<Phi> (a, b) : map\<^sub>0 (src\<^sub>C b) \<rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>C a)\<guillemotright>"
	and "\<guillemotleft>\<Phi> (a, b) : F a \<star>\<^sub>D F b \<Rightarrow>\<^sub>D F (a \<star>\<^sub>C b)\<guillemotright>"
	proof -
	show "\<guillemotleft>\<Phi> (a, b) : F a \<star>\<^sub>D F b \<Rightarrow>\<^sub>D F (a \<star>\<^sub>C b)\<guillemotright>"
	using assms C.VV.arr_char C.VV.dom_char C.VV.cod_char FF_def by auto
	thus "\<guillemotleft>\<Phi> (a, b) : map\<^sub>0 (src\<^sub>C b) \<rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>C a)\<guillemotright>"
	using assms D.vconn_implies_hpar by auto
	qed

	lemma cmp_simps [simp]:
	assumes "C.ide f" and "C.ide g" and "src\<^sub>C f = trg\<^sub>C g"
	shows "D.arr (\<Phi> (f, g))"
	and "src\<^sub>D (\<Phi> (f, g)) = src\<^sub>D (F g)" and "trg\<^sub>D (\<Phi> (f, g)) = trg\<^sub>D (F f)"
	and "D.dom (\<Phi> (f, g)) = F f \<star>\<^sub>D F g" and "D.cod (\<Phi> (f, g)) = F (f \<star>\<^sub>C g)"
	using assms cmp_in_hom by simp_all blast+

	lemma cmp_in_hom':
	assumes "C.arr \<mu>" and "C.arr \<nu>" and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>"
	shows "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : map\<^sub>0 (src\<^sub>C \<nu>) \<rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>C \<mu>)\<guillemotright>"
	and "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : F (C.dom \<mu>) \<star>\<^sub>D F (C.dom \<nu>) \<Rightarrow>\<^sub>D F (C.cod \<mu> \<star>\<^sub>C C.cod \<nu>)\<guillemotright>"
	proof -
	show "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : F (C.dom \<mu>) \<star>\<^sub>D F (C.dom \<nu>) \<Rightarrow>\<^sub>D F (C.cod \<mu> \<star>\<^sub>C C.cod \<nu>)\<guillemotright>"
	using assms C.VV.arr_char C.VV.dom_char C.VV.cod_char FF_def by auto
	thus "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : map\<^sub>0 (src\<^sub>C \<nu>) \<rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>C \<mu>)\<guillemotright>"
	using assms D.vconn_implies_hpar by auto
	qed

	lemma cmp_simps':
	assumes "C.arr \<mu>" and "C.arr \<nu>" and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>"
	shows "D.arr (\<Phi> (\<mu>, \<nu>))"
	and "src\<^sub>D (\<Phi> (\<mu>, \<nu>)) = map\<^sub>0 (src\<^sub>C \<nu>)" and "trg\<^sub>D (\<Phi> (\<mu>, \<nu>)) = map\<^sub>0 (trg\<^sub>C \<mu>)"
	and "D.dom (\<Phi> (\<mu>, \<nu>)) = F (C.dom \<mu>) \<star>\<^sub>D F (C.dom \<nu>)"
	and "D.cod (\<Phi> (\<mu>, \<nu>)) = F (C.cod \<mu> \<star>\<^sub>C C.cod \<nu>)"
	using assms cmp_in_hom' by simp_all blast+

	lemma cmp_components_are_iso [simp]:
	assumes "C.ide f" and "C.ide g" and "src\<^sub>C f = trg\<^sub>C g"
	shows "D.iso (\<Phi> (f, g))"
	using assms C.VV.ide_char C.VV.arr_char by simp

	lemma weakly_preserves_hcomp:
	assumes "C.ide f" and "C.ide g" and "src\<^sub>C f = trg\<^sub>C g"
	shows "D.isomorphic (F f \<star>\<^sub>D F g) (F (f \<star>\<^sub>C g))"
	using assms D.isomorphic_def by auto

	end

	context pseudofunctor
	begin

	text \<open>
	The following defines the image of the unit isomorphism \<open>\<i>\<^sub>C[a]\<close> under \<open>F\<close>.
	We will use \<open>(F a, \<i>[a])\<close> as an ``alternate unit'', to substitute for
	\<open>(src\<^sub>D (F a), \<i>\<^sub>D[src\<^sub>D (F a)])\<close>.
	\<close>

	abbreviation (input) \<i> ("\<i>[_]")
	where "\<i>[a] \<equiv> F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a)"

	lemma \<i>_in_hom [intro]:
	assumes "C.obj a"
	shows "\<guillemotleft>F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a) : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a\<guillemotright>"
	and "\<guillemotleft>\<i>[a] : F a \<star>\<^sub>D F a \<Rightarrow>\<^sub>D F a\<guillemotright>"
	proof (unfold map\<^sub>0_def)
	show "\<guillemotleft>F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a) : F a \<star>\<^sub>D F a \<Rightarrow>\<^sub>D F a\<guillemotright>"
	using assms preserves_hom cmp_in_hom
	by (intro D.comp_in_homI, auto)
	show "\<guillemotleft>F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a) : src\<^sub>D (F a) \<rightarrow>\<^sub>D src\<^sub>D (F a)\<guillemotright>"
	using assms C.VV.arr_char C.VV.dom_simp C.VV.cod_simp
	by (intro D.vcomp_in_hhom D.seqI, auto)
	qed

	lemma \<i>_simps [simp]:
	assumes "C.obj a"
	shows "D.arr (\<i> a)"
	and "src\<^sub>D \<i>[a] = map\<^sub>0 a" and "trg\<^sub>D \<i>[a] = map\<^sub>0 a"
	and "D.dom \<i>[a] = F a \<star>\<^sub>D F a" and "D.cod \<i>[a] = F a"
	using assms \<i>_in_hom by auto

	lemma iso_\<i>:
	assumes "C.obj a"
	shows "D.iso \<i>[a]"
	using assms C.iso_unit C.obj_self_composable(1) C.seq_if_composable
	by (meson C.objE D.isos_compose \<i>_simps(1) cmp_components_are_iso preserves_iso)

	text \<open>
	If \<open>a\<close> is an object of \<open>C\<close> and we have an isomorphism \<open>\<guillemotleft>\<Phi> (a, a) : F a \<star>\<^sub>D F a \<Rightarrow>\<^sub>D F (a \<star>\<^sub>C a)\<guillemotright>\<close>,
	then there is a canonical way to define a compatible isomorphism \<open>\<guillemotleft>\<Psi> a : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>\<close>.
	Specifically, we take \<open>\<Psi> a\<close> to be the unique isomorphism \<open>\<guillemotleft>\<psi> : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>\<close> such that
	\<open>\<psi> \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = \<i>[a] \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)\<close>.
	\<close>

	definition unit
	where "unit a \<equiv> THE \<psi>. \<guillemotleft>\<psi> : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright> \<and> D.iso \<psi> \<and>
	\<psi> \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = \<i>[a] \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"

	lemma unit_char:
	assumes "C.obj a"
	shows "\<guillemotleft>unit a : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>" and "D.iso (unit a)"
	and "unit a \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = \<i>[a] \<cdot>\<^sub>D (unit a \<star>\<^sub>D unit a)"
	and "\<exists>!\<psi>. \<guillemotleft>\<psi> : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = \<i>[a] \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
	proof -
	let ?P = "\<lambda>\<psi>. \<guillemotleft>\<psi> : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = \<i>[a] \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
	show "\<exists>!\<psi>. ?P \<psi>"
	proof -
	have "D.obj (map\<^sub>0 a)"
	using assms by simp
	moreover have "D.isomorphic (map\<^sub>0 a) (F a)"
	unfolding map\<^sub>0_def
	using assms isomorphic_src by simp
	ultimately show ?thesis
	using assms D.unit_unique_upto_unique_iso \<Phi>.preserves_hom \<i>_in_hom iso_\<i> by simp
	qed
	hence 1: "?P (unit a)"
	using assms unit_def the1I2 [of ?P ?P] by simp
	show "\<guillemotleft>unit a : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>" using 1 by simp
	show "D.iso (unit a)" using 1 by simp
	show "unit a \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = \<i>[a] \<cdot>\<^sub>D (unit a \<star>\<^sub>D unit a)"
	using 1 by simp
	qed

	lemma unit_simps [simp]:
	assumes "C.obj a"
	shows "D.arr (unit a)"
	and "src\<^sub>D (unit a) = map\<^sub>0 a" and "trg\<^sub>D (unit a) = map\<^sub>0 a"
	and "D.dom (unit a) = map\<^sub>0 a" and "D.cod (unit a) = F a"
	using assms unit_char(1)
	apply auto
	apply (metis D.vconn_implies_hpar(1) map\<^sub>0_simps(2))
	by (metis D.vconn_implies_hpar(2) map\<^sub>0_simps(3))

	lemma unit_in_hom [intro]:
	assumes "C.obj a"
	shows "\<guillemotleft>unit a : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a\<guillemotright>"
	and "\<guillemotleft>unit a : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>"
	using assms by auto

	lemma unit_eqI:
	assumes "C.obj a" and "\<guillemotleft>\<mu>: map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>" and "D.iso \<mu>"
	and "\<mu> \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = \<i> a \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D \<mu>)"
	shows "\<mu> = unit a"
	using assms unit_def unit_char
	the1_equality [of "\<lambda>\<mu>. \<guillemotleft>\<mu> : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright> \<and> D.iso \<mu> \<and>
	\<mu> \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = \<i>[a] \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D \<mu>)" \<mu>]
	by simp

	text \<open>
	The following defines the unique isomorphism satisfying the characteristic conditions
	for the left unitor \<open>\<l>\<^sub>D[trg\<^sub>D (F f)]\<close>, but using the ``alternate unit'' \<open>\<i>[trg\<^sub>C f]\<close>
	instead of \<open>\<i>\<^sub>D[trg\<^sub>D (F f)]\<close>, which is used to define \<open>\<l>\<^sub>D[trg\<^sub>D (F f)]\<close>.
	\<close>

	definition lF
	where "lF f \<equiv> THE \<mu>. \<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<and>
	F (trg\<^sub>C f) \<star>\<^sub>D \<mu> =(\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"

	lemma lF_char:
	assumes "C.ide f"
	shows "\<guillemotleft>lF f : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright>"
	and "F (trg\<^sub>C f) \<star>\<^sub>D lF f = (\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
	and "\<exists>!\<mu>. \<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<and>
	F (trg\<^sub>C f) \<star>\<^sub>D \<mu> = (\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
	proof -
	let ?P = "\<lambda>\<mu>. \<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<and>
	F (trg\<^sub>C f) \<star>\<^sub>D \<mu> = (\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
	show "\<exists>!\<mu>. ?P \<mu>"
	proof -
	interpret Df: prebicategory \<open>(\<cdot>\<^sub>D)\<close> \<open>(\<star>\<^sub>D)\<close> \<a>\<^sub>D
	using D.is_prebicategory by simp
	interpret S: subcategory \<open>(\<cdot>\<^sub>D)\<close> \<open>Df.left (F (trg\<^sub>C f))\<close>
	using assms Df.left_hom_is_subcategory by simp
	interpret Df: left_hom \<open>(\<cdot>\<^sub>D)\<close> \<open>(\<star>\<^sub>D)\<close> \<open>F (trg\<^sub>C f)\<close>
	using assms D.weak_unit_char
	by unfold_locales simp
	interpret Df: left_hom_with_unit \<open>(\<cdot>\<^sub>D)\<close> \<open>(\<star>\<^sub>D)\<close> \<a>\<^sub>D \<open>\<i>[trg\<^sub>C f]\<close> \<open>F (trg\<^sub>C f)\<close>
	using assms \<i>_in_hom iso_\<i> D.weak_unit_char(1) assms weakly_preserves_trg
	by unfold_locales auto
	have "\<exists>!\<mu>. \<guillemotleft>\<mu> : Df.L (F f) \<Rightarrow>\<^sub>S F f\<guillemotright> \<and>
	Df.L \<mu> = (\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>S \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
	proof -
	have "Df.left (F (trg\<^sub>C f)) (F f)"
	using assms weakly_preserves_src D.isomorphic_def D.hseq_char D.hseq_char'
	Df.left_def
	by fastforce
	thus ?thesis
	using assms Df.lunit_char(3) S.ide_char S.arr_char by simp
	qed
	moreover have "Df.L (F f) = F (trg\<^sub>C f) \<star>\<^sub>D F f"
	using assms by (simp add: Df.H\<^sub>L_def)
	moreover have "\<And>\<mu>. Df.L \<mu> = F (trg\<^sub>C f) \<star>\<^sub>D \<mu>"
	using Df.H\<^sub>L_def by simp
	moreover have "(\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>S \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f] =
	(\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
	by (metis (no_types, lifting) D.arrI D.ext D.hseqI' D.hseq_char' D.seqE
	D.seq_if_composable D.vconn_implies_hpar(1) D.vconn_implies_hpar(2-3)
	D.vconn_implies_hpar(4) Df.\<iota>_in_hom Df.arr_\<omega> S.comp_char S.in_hom_char
	calculation(1,3))
	moreover have "\<And>\<mu>. \<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<longleftrightarrow>
	\<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>S F f\<guillemotright>"
	using assms S.in_hom_char S.arr_char
	by (metis D.in_homE Df.hom_connected(2) Df.left_def calculation(1-2))
	ultimately show ?thesis by simp
	qed
	hence 1: "?P (lF f)"
	using lF_def the1I2 [of ?P ?P] by simp
	show "\<guillemotleft>lF f : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright>"
	using 1 by simp
	show "F (trg\<^sub>C f) \<star>\<^sub>D lF f = (\<i>[trg\<^sub>C f] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
	using 1 by simp
	qed

	lemma lF_simps [simp]:
	assumes "C.ide f"
	shows "D.arr (lF f)"
	and "src\<^sub>D (lF f) = map\<^sub>0 (src\<^sub>C f)" and "trg\<^sub>D (lF f) = map\<^sub>0 (trg\<^sub>C f)"
	and "D.dom (lF f) = F (trg\<^sub>C f) \<star>\<^sub>D F f" and "D.cod (lF f) = F f"
	using assms lF_char(1)
	apply auto[5]
	unfolding map\<^sub>0_def
	using assms
	apply (metis C.ideD(1) D.vconn_implies_hpar(1,3) map\<^sub>0_def preserves_src)
	by (metis C.ideD(1) C.src_trg C.trg.preserves_arr D.in_homE D.trg_cod
	preserves_src preserves_trg)

	text \<open>
	\sloppypar
	The next two lemmas generalize the eponymous results from
	@{theory MonoidalCategory.MonoidalFunctor}. See the proofs of those results for diagrams.
	\<close>

	lemma lunit_coherence1:
	assumes "C.ide f"
	shows "\<l>\<^sub>D[F f] \<cdot>\<^sub>D D.inv (unit (trg\<^sub>C f) \<star>\<^sub>D F f) = lF f"
	proof -
	let ?b = "trg\<^sub>C f"
	have 1: "trg\<^sub>D (F f) = map\<^sub>0 ?b"
	using assms by simp
	have "lF f \<cdot>\<^sub>D (unit ?b \<star>\<^sub>D F f) = \<l>\<^sub>D[F f]"
	proof -
	have "D.par (lF f \<cdot>\<^sub>D (unit ?b \<star>\<^sub>D F f)) \<l>\<^sub>D[F f]"
	using assms 1 D.lunit_in_hom unit_char(1-2) lF_char(1) D.ideD(1) by auto
	moreover have "map\<^sub>0 ?b \<star>\<^sub>D (lF f \<cdot>\<^sub>D (unit ?b \<star>\<^sub>D F f)) = map\<^sub>0 ?b \<star>\<^sub>D \<l>\<^sub>D[F f]"
	proof -
	have "map\<^sub>0 ?b \<star>\<^sub>D (lF f \<cdot>\<^sub>D (unit ?b \<star>\<^sub>D F f)) =
	(map\<^sub>0 ?b \<star>\<^sub>D lF f) \<cdot>\<^sub>D (map\<^sub>0 ?b \<star>\<^sub>D unit ?b \<star>\<^sub>D F f)"
	using assms D.objE [of "map\<^sub>0 (trg\<^sub>C f)"]
	D.whisker_left [of "map\<^sub>0 ?b" "lF f" "unit ?b \<star>\<^sub>D F f"]
	by auto
	also have "... = (map\<^sub>0 ?b \<star>\<^sub>D lF f) \<cdot>\<^sub>D
	(D.inv (unit ?b) \<star>\<^sub>D F ?b \<star>\<^sub>D F f) \<cdot>\<^sub>D (unit ?b \<star>\<^sub>D unit ?b \<star>\<^sub>D F f)"
	proof -
	have "(D.inv (unit ?b) \<star>\<^sub>D F ?b \<star>\<^sub>D F f) \<cdot>\<^sub>D (unit ?b \<star>\<^sub>D unit ?b \<star>\<^sub>D F f) =
	D.inv (unit ?b) \<cdot>\<^sub>D unit ?b \<star>\<^sub>D F ?b \<cdot>\<^sub>D unit ?b \<star>\<^sub>D F f \<cdot>\<^sub>D F f"
	using assms unit_char(1-2)
	D.interchange [of "F ?b" "unit ?b" "F f" "F f"]
	D.interchange [of "D.inv (unit ?b)" "unit ?b" "F ?b \<star>\<^sub>D F f" "unit ?b \<star>\<^sub>D F f"]
	by simp
	also have "... = map\<^sub>0 ?b \<star>\<^sub>D unit ?b \<star>\<^sub>D F f"
	using assms unit_char(1-2) [of ?b] D.comp_arr_dom D.comp_cod_arr D.comp_inv_arr
	by (simp add: D.inv_is_inverse)
	finally show ?thesis by simp
	qed
	also have "... =
	(D.inv (unit ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D (F ?b \<star>\<^sub>D lF f) \<cdot>\<^sub>D (unit ?b \<star>\<^sub>D unit ?b \<star>\<^sub>D F f)"
	proof -
	have "(map\<^sub>0 ?b \<star>\<^sub>D lF f) \<cdot>\<^sub>D (D.inv (unit ?b) \<star>\<^sub>D F ?b \<star>\<^sub>D F f) =
	(D.inv (unit ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D (F ?b \<star>\<^sub>D lF f)"
	proof -
	have "(map\<^sub>0 ?b \<star>\<^sub>D lF f) \<cdot>\<^sub>D (D.inv (unit ?b) \<star>\<^sub>D F ?b \<star>\<^sub>D F f) =
	D.inv (unit ?b) \<star>\<^sub>D lF f"
	using assms unit_char(1-2) lF_char(1) D.comp_arr_dom D.comp_cod_arr
	D.interchange [of "map\<^sub>0 ?b" "D.inv (unit ?b)" "lF f" "F ?b \<star>\<^sub>D F f"]
	by simp
	also have "... = D.inv (unit ?b) \<cdot>\<^sub>D F ?b \<star>\<^sub>D F f \<cdot>\<^sub>D lF f"
	using assms unit_char(1-2) lF_char(1) D.comp_arr_dom D.comp_cod_arr
	D.inv_in_hom
	by auto
	also have "... = (D.inv (unit ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D (F ?b \<star>\<^sub>D lF f)"
	using assms unit_char(1-2) lF_char(1) D.inv_in_hom
	D.interchange [of "D.inv (unit ?b)" "F ?b" "F f" "lF f"]
	by simp
	finally show ?thesis by simp
	qed
	thus ?thesis
	using assms unit_char(1-2) D.inv_in_hom D.comp_assoc by metis
	qed
	also have "... = (D.inv (unit ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<i> ?b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F ?b, F ?b, F f] \<cdot>\<^sub>D
	(unit ?b \<star>\<^sub>D unit ?b \<star>\<^sub>D F f)"
	using assms unit_char(1-2) lF_char(2) D.comp_assoc by auto
	also have "... = ((D.inv (unit ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<i> ?b \<star>\<^sub>D F f) \<cdot>\<^sub>D
	((unit ?b \<star>\<^sub>D unit ?b) \<star>\<^sub>D F f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 ?b, map\<^sub>0 ?b, F f]"
	using assms unit_char(1-2) D.assoc'_naturality [of "unit ?b" "unit ?b" "F f"] D.comp_assoc
	by (simp add: \<open>trg\<^sub>D (F f) = map\<^sub>0 (trg\<^sub>C f)\<close>)
	also have "... = (\<i>\<^sub>D[map\<^sub>0 ?b] \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[map\<^sub>0 ?b, map\<^sub>0 ?b, F f]"
	proof -
	have "((D.inv (unit ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<i> ?b \<star>\<^sub>D F f) \<cdot>\<^sub>D ((unit ?b \<star>\<^sub>D unit ?b) \<star>\<^sub>D F f)) =
	\<i>\<^sub>D[map\<^sub>0 ?b] \<star>\<^sub>D F f"
	proof -
	have "((D.inv (unit ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<i> ?b \<star>\<^sub>D F f) \<cdot>\<^sub>D ((unit ?b \<star>\<^sub>D unit ?b) \<star>\<^sub>D F f)) =
	D.inv (unit ?b) \<cdot>\<^sub>D unit ?b \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 ?b] \<star>\<^sub>D F f"
	using assms 1 D.unit_in_hom D.whisker_right [of "F f"] unit_char(2-3)
	D.invert_side_of_triangle(1)
	by (metis C.ideD(1) C.obj_trg D.seqI' map\<^sub>0_simps(1) unit_in_hom(2) preserves_ide)
	also have "... = \<i>\<^sub>D[map\<^sub>0 ?b] \<star>\<^sub>D F f"
	proof -
	have "(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D unit (trg\<^sub>C f)) \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 ?b] = \<i>\<^sub>D[map\<^sub>0 ?b]"
	by (simp add: D.comp_cod_arr D.comp_inv_arr D.inv_is_inverse unit_char(2)
	assms)
	thus ?thesis
	by (simp add: D.comp_assoc)
	qed
	finally show ?thesis by blast
	qed
	thus ?thesis by simp
	qed
	also have "... = map\<^sub>0 ?b \<star>\<^sub>D \<l>\<^sub>D[F f]"
	using assms D.lunit_char [of "F f"] \<open>trg\<^sub>D (F f) = map\<^sub>0 ?b\<close> by simp
	finally show ?thesis by blast
	qed
	ultimately show ?thesis
	using assms D.L.is_faithful
	by (metis D.trg_cod D.trg_vcomp D.vseq_implies_hpar(2) lF_simps(3))
	qed
	thus ?thesis
	using assms 1 unit_char(1-2) C.ideD(1) C.obj_trg D.inverse_arrows_hcomp(1)
	D.invert_side_of_triangle(2) D.lunit_simps(1) unit_simps(2) preserves_ide
	D.iso_hcomp as_nat_iso.components_are_iso
	by metis
	qed

	lemma lunit_coherence2:
	assumes "C.ide f"
	shows "lF f = F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f)"
	proof -
	let ?b = "trg\<^sub>C f"
	have "D.par (F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (?b, f)) (lF f)"
	using assms cmp_simps'(1) cmp_simps(4-5) by force
	moreover have "F ?b \<star>\<^sub>D F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (?b, f) = F ?b \<star>\<^sub>D lF f"
	proof -
	have "F ?b \<star>\<^sub>D F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (?b, f) = (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D (F ?b \<star>\<^sub>D \<Phi> (?b, f))"
	using assms cmp_in_hom D.whisker_left [of "F ?b" "F \<l>\<^sub>C[f]" "\<Phi> (?b, f)"]
	by (simp add: calculation)
	also have "... = F ?b \<star>\<^sub>D lF f"
	proof -
	have "(F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D (F ?b \<star>\<^sub>D \<Phi> (?b, f))
	= (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D F \<a>\<^sub>C[?b, ?b, f] \<cdot>\<^sub>D
	\<Phi> (?b \<star>\<^sub>C ?b, f) \<cdot>\<^sub>D (\<Phi> (?b, ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F ?b, F ?b, F f]"
	proof -
	have 1: "D.seq (F \<a>\<^sub>C[trg\<^sub>C f, trg\<^sub>C f, f])
	(\<Phi> (trg\<^sub>C f \<star>\<^sub>C trg\<^sub>C f, f) \<cdot>\<^sub>D (\<Phi> (trg\<^sub>C f, trg\<^sub>C f) \<star>\<^sub>D F f))"
	using assms by fastforce
	hence 2: "D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D F \<a>\<^sub>C[?b, ?b, f] \<cdot>\<^sub>D \<Phi> (?b \<star>\<^sub>C ?b, f) \<cdot>\<^sub>D
	(\<Phi> (?b, ?b) \<star>\<^sub>D F f)
	= (F ?b \<star>\<^sub>D \<Phi> (?b, f)) \<cdot>\<^sub>D \<a>\<^sub>D[F ?b, F ?b, F f]"
	using assms cmp_in_hom assoc_coherence cmp_components_are_iso
	D.invert_side_of_triangle(1)
	[of "F \<a>\<^sub>C[?b, ?b, f] \<cdot>\<^sub>D \<Phi> (?b \<star>\<^sub>C ?b, f) \<cdot>\<^sub>D (\<Phi> (?b, ?b) \<star>\<^sub>D F f)"
	"\<Phi> (?b, ?b \<star>\<^sub>C f)"
	"(F ?b \<star>\<^sub>D \<Phi> (?b, f)) \<cdot>\<^sub>D \<a>\<^sub>D[F ?b, F ?b, F f]"]
	C.ideD(1) C.ide_hcomp C.trg_hcomp C.trg_trg C.src_trg C.trg.preserves_ide
	by metis
	hence "F ?b \<star>\<^sub>D \<Phi> (?b, f)
	= (D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D F \<a>\<^sub>C[?b, ?b, f] \<cdot>\<^sub>D \<Phi> (?b \<star>\<^sub>C ?b, f) \<cdot>\<^sub>D
	(\<Phi> (?b, ?b) \<star>\<^sub>D F f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F ?b, F ?b, F f]"
	proof -
	have "D.seq (D.inv (\<Phi> (trg\<^sub>C f, trg\<^sub>C f \<star>\<^sub>C f)))
	(F \<a>\<^sub>C[trg\<^sub>C f, trg\<^sub>C f, f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f \<star>\<^sub>C trg\<^sub>C f, f) \<cdot>\<^sub>D
	(\<Phi> (trg\<^sub>C f, trg\<^sub>C f) \<star>\<^sub>D F f))"
	using assms 1 D.hseq_char by auto
	moreover have "(F (trg\<^sub>C f) \<star>\<^sub>D \<Phi> (trg\<^sub>C f, f)) \<cdot>\<^sub>D \<a>\<^sub>D[F (trg\<^sub>C f), F (trg\<^sub>C f), F f] =
	D.inv (\<Phi> (trg\<^sub>C f, trg\<^sub>C f \<star>\<^sub>C f)) \<cdot>\<^sub>D
	F \<a>\<^sub>C[trg\<^sub>C f, trg\<^sub>C f, f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f \<star>\<^sub>C trg\<^sub>C f, f) \<cdot>\<^sub>D
	(\<Phi> (trg\<^sub>C f, trg\<^sub>C f) \<star>\<^sub>D F f)"
	using assms 2 by simp
	ultimately show ?thesis
	using assms
	D.invert_side_of_triangle(2)
	[of "D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D F \<a>\<^sub>C[?b, ?b, f] \<cdot>\<^sub>D \<Phi> (?b \<star>\<^sub>C ?b, f) \<cdot>\<^sub>D
	(\<Phi> (?b, ?b) \<star>\<^sub>D F f)"
	"F ?b \<star>\<^sub>D \<Phi> (?b, f)" "\<a>\<^sub>D[F ?b, F ?b, F f]"]
	by fastforce
	qed
	thus ?thesis
	using D.comp_assoc by simp
	qed
	also have "... = (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D
	(D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D F (\<i>\<^sub>C[?b] \<star>\<^sub>C f)) \<cdot>\<^sub>D
	\<Phi> (?b \<star>\<^sub>C ?b, f) \<cdot>\<^sub>D (\<Phi> (?b, ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D
	\<a>\<^sub>D\<^sup>-\<^sup>1[F ?b, F ?b, F f]"
	proof -
	have 1: "F (?b \<star>\<^sub>C \<l>\<^sub>C[f]) = F (\<i>\<^sub>C[?b] \<star>\<^sub>C f) \<cdot>\<^sub>D D.inv (F \<a>\<^sub>C[?b, ?b, f])"
	using assms C.lunit_char(1-2) C.unit_in_hom preserves_inv by auto
	have "F \<a>\<^sub>C[?b, ?b, f] = D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D F (\<i>\<^sub>C[?b] \<star>\<^sub>C f)"
	proof -
	have "F \<a>\<^sub>C[?b, ?b, f] \<cdot>\<^sub>D D.inv (F (\<i>\<^sub>C[?b] \<star>\<^sub>C f)) =
	D.inv (F (\<i>\<^sub>C[?b] \<star>\<^sub>C f) \<cdot>\<^sub>D D.inv (F \<a>\<^sub>C[?b, ?b, f]))"
	using assms by (simp add: C.iso_unit D.inv_comp)
	thus ?thesis
	using assms 1 D.invert_side_of_triangle D.iso_inv_iso
	by (metis C.iso_hcomp C.ideD(1) C.ide_is_iso C.iso_lunit C.iso_unit
	C.lunit_simps(3) C.obj_trg C.src_trg C.trg.as_nat_iso.components_are_iso
	C.unit_simps(2) D.arr_inv D.inv_inv preserves_iso)
	qed
	thus ?thesis by argo
	qed
	also have "... = (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D
	D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D (F (\<i>\<^sub>C[?b] \<star>\<^sub>C f) \<cdot>\<^sub>D \<Phi> (?b \<star>\<^sub>C ?b, f)) \<cdot>\<^sub>D
	(\<Phi> (?b, ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F ?b, F ?b, F f]"
	using D.comp_assoc by auto
	also have "... = (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D
	D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D (\<Phi> (?b, f) \<cdot>\<^sub>D (F \<i>\<^sub>C[?b] \<star>\<^sub>D F f)) \<cdot>\<^sub>D
	(\<Phi> (?b, ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F ?b, F ?b, F f]"
	using assms \<Phi>.naturality [of "(\<i>\<^sub>C[?b], f)"] FF_def C.VV.arr_char C.VV.cod_char
	C.VV.dom_char
	by simp
	also have "... = (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D
	D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D \<Phi> (?b, f) \<cdot>\<^sub>D
	((F \<i>\<^sub>C[?b] \<star>\<^sub>D F f)) \<cdot>\<^sub>D (\<Phi> (?b, ?b) \<star>\<^sub>D F f) \<cdot>\<^sub>D
	\<a>\<^sub>D\<^sup>-\<^sup>1[F ?b, F ?b, F f]"
	using D.comp_assoc by auto
	also have "... = (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D
	D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D \<Phi> (?b, f) \<cdot>\<^sub>D (\<i> ?b \<star>\<^sub>D F f) \<cdot>\<^sub>D
	\<a>\<^sub>D\<^sup>-\<^sup>1[F ?b, F ?b, F f]"
	using assms by (simp add: D.comp_assoc D.whisker_right)
	also have "... = (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D
	(D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D \<Phi> (?b, f)) \<cdot>\<^sub>D
	(F ?b \<star>\<^sub>D lF f)"
	using D.comp_assoc assms lF_char(2) by presburger
	also have "... = (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D (F ?b \<star>\<^sub>D lF f)"
	proof -
	have "D.inv (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) =
	D.inv (((F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f))) \<cdot>\<^sub>D \<Phi> (?b, ?b \<star>\<^sub>C f))"
	using assms D.comp_inv_arr D.comp_inv_arr' cmp_simps(4)
	D.comp_arr_dom D.comp_assoc
	by simp
	also have "... = D.inv (D.inv (\<Phi> (?b, f)) \<cdot>\<^sub>D F (?b \<star>\<^sub>C \<l>\<^sub>C[f]) \<cdot>\<^sub>D \<Phi> (?b, ?b \<star>\<^sub>C f))"
	proof -
	have 1: "\<Phi> (?b, f) \<cdot>\<^sub>D (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) = F (?b \<star>\<^sub>C \<l>\<^sub>C[f]) \<cdot>\<^sub>D \<Phi> (?b, ?b \<star>\<^sub>C f)"
	using assms \<Phi>.naturality [of "(?b, \<l>\<^sub>C[f])"] FF_def C.VV.arr_char
	C.VV.cod_char D.VV.null_char C.VV.dom_simp
	by simp
	have "(F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) =
	D.inv (\<Phi> (?b, f)) \<cdot>\<^sub>D F (?b \<star>\<^sub>C \<l>\<^sub>C[f])"
	proof -
	have "D.seq (\<Phi> (?b, f)) (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f])"
	using assms cmp_in_hom(2) [of ?b f] by auto
	moreover have "D.iso (\<Phi> (?b, f)) \<and> D.iso (\<Phi> (?b, ?b \<star>\<^sub>C f))"
	using assms by simp
	ultimately show ?thesis
	using 1 D.invert_opposite_sides_of_square by simp
	qed
	thus ?thesis
	using D.comp_assoc by auto
	qed
	also have "... = D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f]) \<cdot>\<^sub>D \<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D \<Phi> (?b, f)"
	proof -
	have "D.iso (F (?b \<star>\<^sub>C \<l>\<^sub>C[f]) \<cdot>\<^sub>D \<Phi> (?b, ?b \<star>\<^sub>C f))"
	using assms D.isos_compose C.VV.arr_char C.iso_lunit C.VV.dom_simp
	C.VV.cod_simp
	by simp
	moreover have "D.iso (D.inv (\<Phi> (?b, f)))"
	using assms by simp
	moreover have "D.seq (D.inv (\<Phi> (?b, f))) (F (?b \<star>\<^sub>C \<l>\<^sub>C[f]) \<cdot>\<^sub>D \<Phi> (?b, ?b \<star>\<^sub>C f))"
	using assms C.VV.arr_char C.VV.dom_simp C.VV.cod_simp by simp
	ultimately show ?thesis
	using assms D.inv_comp by simp
	qed
	also have "... = D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D \<Phi> (?b, f)"
	using D.comp_assoc D.inv_comp assms cmp_simps'(1) cmp_simps(5) by force
	finally have "D.inv (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f])
	= D.inv (\<Phi> (?b, ?b \<star>\<^sub>C f)) \<cdot>\<^sub>D D.inv (F (?b \<star>\<^sub>C \<l>\<^sub>C[f])) \<cdot>\<^sub>D \<Phi> (?b, f)"
	by blast
	thus ?thesis by argo
	qed
	also have "... = ((F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D D.inv (F ?b \<star>\<^sub>D F \<l>\<^sub>C[f])) \<cdot>\<^sub>D (F ?b \<star>\<^sub>D lF f)"
	using D.comp_assoc by simp
	also have "... = F ?b \<star>\<^sub>D lF f"
	using assms D.comp_arr_inv' [of "F ?b \<star>\<^sub>D F \<l>\<^sub>C[f]"] D.comp_cod_arr by simp
	finally show ?thesis by simp
	qed
	ultimately show ?thesis by simp
	qed
	ultimately show ?thesis
	using assms D.L.is_faithful
	by (metis D.in_homI lF_char(2-3) lF_simps(4-5))
	qed

	lemma lunit_coherence:
	assumes "C.ide f"
	shows "\<l>\<^sub>D[F f] = F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)"
	proof -
	have "\<l>\<^sub>D[F f] = (F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f)) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)"
	by (metis C.ideD(1) C.obj_trg D.inv_inv D.invert_side_of_triangle(2)
	D.iso_hcomp D.iso_inv_iso as_nat_iso.components_are_iso assms lF_simps(1)
	lunit_coherence1 lunit_coherence2 preserves_trg unit_char(2) unit_simps(2))
	thus ?thesis
	using assms D.comp_assoc by simp
	qed

	text \<open>
	We postpone proving the dual version of this result until after we have developed
	the notion of the ``op bicategory'' in the next section.
	\<close>

	end

	subsection "Pseudofunctors and Opposite Bicategories"

	text \<open>
	There are three duals to a bicategory:
	\begin{enumerate}
	\item ``op'': sources and targets are exchanged;
	\item ``co'': domains and codomains are exchanged;
	\item ``co-op'': both sources and targets and domains and codomains are exchanged.
	\end{enumerate}
	Here we consider the "op" case.
	\<close>

	locale op_bicategory =
	B: bicategory V H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
	for V :: "'a comp" (infixr "\<cdot>" 55)
	and H\<^sub>B :: "'a comp" (infixr "\<star>\<^sub>B" 53)
	and \<a>\<^sub>B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("\<a>\<^sub>B[_, _, _]")
	and \<i>\<^sub>B :: "'a \<Rightarrow> 'a" ("\<i>\<^sub>B[_]")
	and src\<^sub>B :: "'a \<Rightarrow> 'a"
	and trg\<^sub>B :: "'a \<Rightarrow> 'a"
	begin

	abbreviation H (infixr "\<star>" 53)
	where "H f g \<equiv> H\<^sub>B g f"

	abbreviation \<i> ("\<i>[_]")
	where "\<i> \<equiv> \<i>\<^sub>B"

	abbreviation src
	where "src \<equiv> trg\<^sub>B"

	abbreviation trg
	where "trg \<equiv> src\<^sub>B"

	interpretation horizontal_homs V src trg
	by (unfold_locales, auto)

	interpretation H: "functor" VV.comp V \<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star> snd \<mu>\<nu>\<close>
	using VV.arr_char VV.dom_simp VV.cod_simp
	apply unfold_locales
	apply (metis (no_types, lifting) B.hseqE B.hseq_char')
	apply auto[3]
	using VV.comp_char VV.seq_char VV.arr_char B.VxV.comp_char B.interchange
	by (metis (no_types, lifting) B.VxV.seqE fst_conv snd_conv)

	interpretation horizontal_composition V H src trg
	by (unfold_locales, auto)

	abbreviation UP
	where "UP \<mu>\<nu>\<tau> \<equiv> if B.VVV.arr \<mu>\<nu>\<tau> then
	(snd (snd \<mu>\<nu>\<tau>), fst (snd \<mu>\<nu>\<tau>), fst \<mu>\<nu>\<tau>)
	else VVV.null"

	abbreviation DN
	where "DN \<mu>\<nu>\<tau> \<equiv> if VVV.arr \<mu>\<nu>\<tau> then
	(snd (snd \<mu>\<nu>\<tau>), fst (snd \<mu>\<nu>\<tau>), fst \<mu>\<nu>\<tau>)
	else B.VVV.null"

	lemma VVV_arr_char:
	shows "VVV.arr \<mu>\<nu>\<tau> \<longleftrightarrow> B.VVV.arr (DN \<mu>\<nu>\<tau>)"
	using VVV.arr_char VV.arr_char B.VVV.arr_char B.VV.arr_char B.VVV.not_arr_null
	by auto

	lemma VVV_ide_char:
	shows "VVV.ide \<mu>\<nu>\<tau> \<longleftrightarrow> B.VVV.ide (DN \<mu>\<nu>\<tau>)"
	proof -
	have "VVV.ide \<mu>\<nu>\<tau> \<longleftrightarrow> VVV.arr \<mu>\<nu>\<tau> \<and> B.VxVxV.ide \<mu>\<nu>\<tau>"
	using VVV.ide_char by simp
	also have "... \<longleftrightarrow> B.VVV.arr (DN \<mu>\<nu>\<tau>) \<and> B.VxVxV.ide (DN \<mu>\<nu>\<tau>)"
	using VVV_arr_char B.VxVxV.ide_char by auto
	also have "... \<longleftrightarrow> B.VVV.ide (DN \<mu>\<nu>\<tau>)"
	using B.VVV.ide_char [of "DN \<mu>\<nu>\<tau>"] by blast
	finally show ?thesis by fast
	qed

	lemma VVV_dom_char:
	shows "VVV.dom \<mu>\<nu>\<tau> = UP (B.VVV.dom (DN \<mu>\<nu>\<tau>))"
	proof (cases "VVV.arr \<mu>\<nu>\<tau>")
	show "\<not> VVV.arr \<mu>\<nu>\<tau> \<Longrightarrow> VVV.dom \<mu>\<nu>\<tau> = UP (B.VVV.dom (DN \<mu>\<nu>\<tau>))"
	using VVV.dom_def VVV.has_domain_iff_arr VVV_arr_char B.VVV.dom_null
	by auto
	show "VVV.arr \<mu>\<nu>\<tau> \<Longrightarrow> VVV.dom \<mu>\<nu>\<tau> = UP (B.VVV.dom (DN \<mu>\<nu>\<tau>))"
	proof -
	assume \<mu>\<nu>\<tau>: "VVV.arr \<mu>\<nu>\<tau>"
	have "VVV.dom \<mu>\<nu>\<tau> =
	(B.dom (fst \<mu>\<nu>\<tau>), B.dom (fst (snd \<mu>\<nu>\<tau>)), B.dom (snd (snd \<mu>\<nu>\<tau>)))"
	using \<mu>\<nu>\<tau> VVV.dom_char VVV.arr_char VV.arr_char B.VVV.arr_char B.VV.arr_char
	by simp
	also have "... = UP (B.dom (snd (snd \<mu>\<nu>\<tau>)), B.dom (fst (snd \<mu>\<nu>\<tau>)), B.dom (fst \<mu>\<nu>\<tau>))"
	by (metis (no_types, lifting) B.VV.arrI B.VVV.arr_char B.arr_dom VV.arrE VVV.arrE
	\<mu>\<nu>\<tau> fst_conv snd_conv src_dom trg_dom)
	also have "... = UP (B.VVV.dom (DN \<mu>\<nu>\<tau>))"
	using \<mu>\<nu>\<tau> B.VVV.dom_char B.VVV.arr_char B.VV.arr_char VVV.arr_char VV.arr_char
	by simp
	finally show ?thesis by blast
	qed
	qed

	lemma VVV_cod_char:
	shows "VVV.cod \<mu>\<nu>\<tau> = UP (B.VVV.cod (DN \<mu>\<nu>\<tau>))"
	proof (cases "VVV.arr \<mu>\<nu>\<tau>")
	show "\<not> VVV.arr \<mu>\<nu>\<tau> \<Longrightarrow> VVV.cod \<mu>\<nu>\<tau> = UP (B.VVV.cod (DN \<mu>\<nu>\<tau>))"
	using VVV.cod_def VVV.has_codomain_iff_arr VVV_arr_char B.VVV.cod_null
	by auto
	show "VVV.arr \<mu>\<nu>\<tau> \<Longrightarrow> VVV.cod \<mu>\<nu>\<tau> = UP (B.VVV.cod (DN \<mu>\<nu>\<tau>))"
	proof -
	assume \<mu>\<nu>\<tau>: "VVV.arr \<mu>\<nu>\<tau>"
	have "VVV.cod \<mu>\<nu>\<tau> = (B.cod (fst \<mu>\<nu>\<tau>), B.cod (fst (snd \<mu>\<nu>\<tau>)), B.cod (snd (snd \<mu>\<nu>\<tau>)))"
	using \<mu>\<nu>\<tau> VVV.cod_char VVV.arr_char VV.arr_char B.VVV.arr_char B.VV.arr_char
	by simp
	also have "... = UP (B.cod (snd (snd \<mu>\<nu>\<tau>)), B.cod (fst (snd \<mu>\<nu>\<tau>)), B.cod (fst \<mu>\<nu>\<tau>))"
	by (metis (no_types, lifting) B.VV.arrI B.VVV.arr_char B.arr_cod VV.arrE VVV.arrE
	\<mu>\<nu>\<tau> fst_conv snd_conv src_cod trg_cod)
	also have "... = UP (B.VVV.cod (DN \<mu>\<nu>\<tau>))"
	using \<mu>\<nu>\<tau> B.VVV.cod_char B.VVV.arr_char B.VV.arr_char VVV.arr_char VV.arr_char
	by simp
	finally show ?thesis by blast
	qed
	qed

	lemma HoHV_char:
	shows "HoHV \<mu>\<nu>\<tau> = B.HoVH (DN \<mu>\<nu>\<tau>)"
	using HoHV_def B.HoVH_def VVV_arr_char by simp

	lemma HoVH_char:
	shows "HoVH \<mu>\<nu>\<tau> = B.HoHV (DN \<mu>\<nu>\<tau>)"
	using HoVH_def B.HoHV_def VVV_arr_char by simp

	definition \<a> ("\<a>[_, _, _]")
	where "\<a>[\<mu>, \<nu>, \<tau>] \<equiv> B.\<alpha>' (DN (\<mu>, \<nu>, \<tau>))"

	interpretation natural_isomorphism VVV.comp \<open>(\<cdot>)\<close> HoHV HoVH
	\<open>\<lambda>\<mu>\<nu>\<tau>. \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]\<close>
	proof
	fix \<mu>\<nu>\<tau>
	show "\<not> VVV.arr \<mu>\<nu>\<tau> \<Longrightarrow> \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)] = B.null"
	using VVV.arr_char B.VVV.arr_char \<a>_def B.\<alpha>'.is_extensional by auto
	assume \<mu>\<nu>\<tau>: "VVV.arr \<mu>\<nu>\<tau>"
	show "B.dom \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)] = HoHV (VVV.dom \<mu>\<nu>\<tau>)"
	using \<mu>\<nu>\<tau> \<a>_def HoHV_def B.\<alpha>'.preserves_dom VVV.arr_char VVV.dom_char VV.arr_char
	B.HoVH_def B.VVV.arr_char B.VV.arr_char B.VVV.dom_char
	by auto
	show "B.cod \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)] = HoVH (VVV.cod \<mu>\<nu>\<tau>)"
	using \<mu>\<nu>\<tau> \<a>_def HoVH_def B.\<alpha>'.preserves_cod VVV.arr_char VVV.cod_char VV.arr_char
	B.HoHV_def B.VVV.arr_char B.VV.arr_char B.VVV.cod_char
	by auto
	show "HoVH \<mu>\<nu>\<tau> \<cdot>
	\<a>[fst (VVV.dom \<mu>\<nu>\<tau>), fst (snd (VVV.dom \<mu>\<nu>\<tau>)), snd (snd (VVV.dom \<mu>\<nu>\<tau>))] =
	\<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
	proof -
	have "HoVH \<mu>\<nu>\<tau> \<cdot>
	\<a>[fst (VVV.dom \<mu>\<nu>\<tau>), fst (snd (VVV.dom \<mu>\<nu>\<tau>)), snd (snd (VVV.dom \<mu>\<nu>\<tau>))] =
	HoVH \<mu>\<nu>\<tau> \<cdot> B.\<alpha>' (B.VVV.dom (snd (snd \<mu>\<nu>\<tau>), fst (snd \<mu>\<nu>\<tau>), fst \<mu>\<nu>\<tau>))"
	unfolding \<a>_def
	using \<mu>\<nu>\<tau> VVV.arr_char VV.arr_char VVV.dom_char B.VVV.arr_char B.VVV.dom_char
	by auto
	also have "... = B.\<alpha>' (snd (snd \<mu>\<nu>\<tau>), fst (snd \<mu>\<nu>\<tau>), fst \<mu>\<nu>\<tau>)"
	using B.\<alpha>'.is_natural_1 VVV_arr_char \<mu>\<nu>\<tau> HoVH_char by presburger
	also have "... = \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
	using \<mu>\<nu>\<tau> \<a>_def by simp
	finally show ?thesis by blast
	qed
	show "\<a>[fst (VVV.cod \<mu>\<nu>\<tau>), fst (snd (VVV.cod \<mu>\<nu>\<tau>)), snd (snd (VVV.cod \<mu>\<nu>\<tau>))] \<cdot>
	HoHV \<mu>\<nu>\<tau> =
	\<a> (fst \<mu>\<nu>\<tau>) (fst (snd \<mu>\<nu>\<tau>)) (snd (snd \<mu>\<nu>\<tau>))"
	proof -
	have "\<a>[fst (VVV.cod \<mu>\<nu>\<tau>), fst (snd (VVV.cod \<mu>\<nu>\<tau>)), snd (snd (VVV.cod \<mu>\<nu>\<tau>))] \<cdot>
	HoHV \<mu>\<nu>\<tau> =
	B.\<alpha>' (B.VVV.cod (snd (snd \<mu>\<nu>\<tau>), fst (snd \<mu>\<nu>\<tau>), fst \<mu>\<nu>\<tau>)) \<cdot> HoHV \<mu>\<nu>\<tau>"
	unfolding \<a>_def
	using \<mu>\<nu>\<tau> VVV.arr_char VV.arr_char VVV.cod_char B.VVV.arr_char B.VVV.cod_char
	by auto
	also have "... = B.\<alpha>' (snd (snd \<mu>\<nu>\<tau>), fst (snd \<mu>\<nu>\<tau>), fst \<mu>\<nu>\<tau>)"
	using B.\<alpha>'.is_natural_2 VVV_arr_char \<mu>\<nu>\<tau> HoHV_char by presburger
	also have "... = \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
	using \<mu>\<nu>\<tau> \<a>_def by simp
	finally show ?thesis by blast
	qed
	next
	fix \<mu>\<nu>\<tau>
	assume \<mu>\<nu>\<tau>: "VVV.ide \<mu>\<nu>\<tau>"
	show "B.iso \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
	proof -
	have "B.VVV.ide (DN \<mu>\<nu>\<tau>)"
	using \<mu>\<nu>\<tau> VVV_ide_char by blast
	thus ?thesis
	using \<mu>\<nu>\<tau> \<a>_def B.\<alpha>'.components_are_iso by force
	qed
	qed

	sublocale bicategory V H \<a> \<i> src trg
	proof
	show "\<And>a. obj a \<Longrightarrow> \<guillemotleft>\<i> a : H a a \<rightarrow>\<^sub>B a\<guillemotright>"
	using obj_def objE B.obj_def B.objE B.unit_in_hom by meson
	show "\<And>a. obj a \<Longrightarrow> B.iso (\<i> a)"
	using obj_def objE B.obj_def B.objE B.iso_unit by meson
	show "\<And>f g h k. \<lbrakk> B.ide f; B.ide g; B.ide h; B.ide k;
	src f = trg g; src g = trg h; src h = trg k \<rbrakk> \<Longrightarrow>
	(f \<star> \<a>[g, h, k]) \<cdot> \<a>[f, g \<star> h, k] \<cdot> (\<a>[f, g, h] \<star> k) = \<a>[f, g, h \<star> k] \<cdot> \<a>[f \<star> g, h, k]"
	unfolding \<a>_def
	using B.\<a>'_def B.comp_assoc B.pentagon' VVV.arr_char VV.arr_char by simp
	qed

	proposition is_bicategory:
	shows "bicategory V H \<a> \<i> src trg"
	..

	lemma assoc_ide_simp:
	assumes "B.ide f" and "B.ide g" and "B.ide h"
	and "src f = trg g" and "src g = trg h"
	shows "\<a>[f, g, h] = B.\<a>' h g f"
	using assms \<a>_def B.\<a>'_def by fastforce

	lemma lunit_ide_simp:
	assumes "B.ide f"
	shows "lunit f = B.runit f"
	proof (intro B.runit_eqI)
	show "B.ide f" by fact
	show "\<guillemotleft>lunit f : H (trg f) f \<rightarrow>\<^sub>B f\<guillemotright>"
	using assms by simp
	show "trg f \<star> lunit f = (\<i>[trg f] \<star> f) \<cdot> \<a>\<^sub>B[f, trg f, trg f]"
	proof -
	have "trg f \<star> lunit f = (\<i>[trg f] \<star> f) \<cdot> \<a>' (trg f) (trg f) f"
	using assms lunit_char(1-2) [of f] by simp
	moreover have "\<a>' (trg f) (trg f) f = \<a>\<^sub>B[f, trg f, trg f]"
	proof (unfold \<a>'_def)
	have 1: "VVV.ide (trg f, trg f, f)"
	using assms VVV.ide_char VVV.arr_char VV.arr_char by simp
	have "\<alpha>' (trg f, trg f, f) = B.inv \<a>[trg f, trg f, f]"
	using 1 B.\<alpha>'.inverts_components by simp
	also have "... = B.inv (B.\<alpha>' (f, trg f, trg f))"
	unfolding \<a>_def using 1 by simp
	also have "... = \<a>\<^sub>B[f, trg f, trg f]"
	using 1 B.VVV.ide_char B.VVV.arr_char B.VV.arr_char VVV.ide_char
	VVV.arr_char VV.arr_char B.\<alpha>'.inverts_components B.\<alpha>_def
	by force
	finally show "\<alpha>' (trg f, trg f, f) = \<a>\<^sub>B[f, trg f, trg f]"
	by blast
	qed
	ultimately show ?thesis by simp
	qed
	qed

	lemma runit_ide_simp:
	assumes "B.ide f"
	shows "runit f = B.lunit f"
	using assms runit_char(1-2) [of f] B.\<a>'_def assoc_ide_simp
	by (intro B.lunit_eqI) auto

	end

	context pseudofunctor
	begin

	interpretation C': op_bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C ..
	interpretation D': op_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D ..

	notation C'.H (infixr "\<star>\<^sub>C\<^sup>o\<^sup>p" 53)
	notation D'.H (infixr "\<star>\<^sub>D\<^sup>o\<^sup>p" 53)

	interpretation F': weak_arrow_of_homs V\<^sub>C C'.src C'.trg V\<^sub>D D'.src D'.trg F
	apply unfold_locales
	using weakly_preserves_src weakly_preserves_trg by simp_all
	interpretation H\<^sub>D'oFF: composite_functor C'.VV.comp D'.VV.comp V\<^sub>D F'.FF
	\<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>D\<^sup>o\<^sup>p snd \<mu>\<nu>\<close> ..
	interpretation FoH\<^sub>C': composite_functor C'.VV.comp V\<^sub>C V\<^sub>D
	\<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>C\<^sup>o\<^sup>p snd \<mu>\<nu>\<close> F
	..
	interpretation \<Phi>': natural_isomorphism C'.VV.comp V\<^sub>D H\<^sub>D'oFF.map FoH\<^sub>C'.map
	\<open>\<lambda>f. \<Phi> (snd f, fst f)\<close>
	using C.VV.arr_char C'.VV.arr_char C'.VV.ide_char C.VV.ide_char FF_def F'.FF_def
	\<Phi>.is_extensional \<Phi>.is_natural_1 \<Phi>.is_natural_2
	C.VV.dom_simp C.VV.cod_simp C'.VV.dom_simp C'.VV.cod_simp
	by unfold_locales auto
	interpretation F': pseudofunctor V\<^sub>C C'.H C'.\<a> \<i>\<^sub>C C'.src C'.trg
	V\<^sub>D D'.H D'.\<a> \<i>\<^sub>D D'.src D'.trg
	F \<open>\<lambda>f. \<Phi> (snd f, fst f)\<close>
	proof
	fix f g h
	assume f: "C.ide f" and g: "C.ide g" and h: "C.ide h"
	and fg: "C'.src f = C'.trg g" and gh: "C'.src g = C'.trg h"
	show "F (C'.\<a> f g h) \<cdot>\<^sub>D \<Phi> (snd (f \<star>\<^sub>C\<^sup>o\<^sup>p g, h), fst (f \<star>\<^sub>C\<^sup>o\<^sup>p g, h)) \<cdot>\<^sub>D
	(\<Phi> (snd (f, g), fst (f, g)) \<star>\<^sub>D\<^sup>o\<^sup>p F h) =
	\<Phi> (snd (f, g \<star>\<^sub>C\<^sup>o\<^sup>p h), fst (f, g \<star>\<^sub>C\<^sup>o\<^sup>p h)) \<cdot>\<^sub>D (F f \<star>\<^sub>D\<^sup>o\<^sup>p \<Phi> (snd (g, h), fst (g, h))) \<cdot>\<^sub>D
	D'.\<a> (F f) (F g) (F h)"
	using f g h fg gh C.VV.in_hom_char FF_def C.VV.arr_char C.VV.dom_simp C.VV.cod_simp
	C'.assoc_ide_simp D'.assoc_ide_simp preserves_inv assoc_coherence
	D.invert_opposite_sides_of_square
	[of "F (\<a>\<^sub>C h g f)" "\<Phi> (g \<star>\<^sub>C\<^sup>o\<^sup>p h, f) \<cdot>\<^sub>D (F f \<star>\<^sub>D\<^sup>o\<^sup>p \<Phi> (h, g))"
	"\<Phi> (h, f \<star>\<^sub>C\<^sup>o\<^sup>p g) \<cdot>\<^sub>D (\<Phi> (g, f) \<star>\<^sub>D\<^sup>o\<^sup>p F h)" "\<a>\<^sub>D (F h) (F g) (F f)"]
	D.comp_assoc
	by auto
	qed

	lemma induces_pseudofunctor_between_opposites:
	shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C\<^sup>o\<^sup>p) C'.\<a> \<i>\<^sub>C C'.src C'.trg
	(\<cdot>\<^sub>D) (\<star>\<^sub>D\<^sup>o\<^sup>p) D'.\<a> \<i>\<^sub>D D'.src D'.trg
	F (\<lambda>f. \<Phi> (snd f, fst f))"
	..

	text \<open>
	It is now easy to dualize the coherence condition for \<open>F\<close> with respect to
	left unitors to obtain the corresponding condition for right unitors.
	\<close>

	lemma runit_coherence:
	assumes "C.ide f"
	shows "\<r>\<^sub>D[F f] = F \<r>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))"
	proof -
	have "C'.lunit f = \<r>\<^sub>C[f]"
	using assms C'.lunit_ide_simp by simp
	moreover have "D'.lunit (F f) = \<r>\<^sub>D[F f]"
	using assms D'.lunit_ide_simp by simp
	moreover have "F'.unit (src\<^sub>C f) = unit (src\<^sub>C f)"
	using assms F'.unit_char F'.preserves_trg
	by (intro unit_eqI) auto
	moreover have "D'.lunit (F f) =
	F (C'.lunit f) \<cdot>\<^sub>D \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D F'.unit (src\<^sub>C f))"
	using assms F'.lunit_coherence by simp
	ultimately show ?thesis by simp
	qed

	end

	subsection "Preservation Properties"

	text \<open>
	The objective of this section is to establish explicit formulas for the result
	of applying a pseudofunctor to expressions of various forms.
	\<close>

	context pseudofunctor
	begin

	lemma preserves_lunit:
	assumes "C.ide f"
	shows "F \<l>\<^sub>C[f] = \<l>\<^sub>D[F f] \<cdot>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f))"
	and "F \<l>\<^sub>C\<^sup>-\<^sup>1[f] = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f]"
	proof -
	show 1: "F \<l>\<^sub>C[f] = \<l>\<^sub>D[F f] \<cdot>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f))"
	proof -
	have "\<l>\<^sub>D[F f] \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)) = F \<l>\<^sub>C[f]"
	proof -
	have "D.arr \<l>\<^sub>D[F f]"
	using assms by simp
	moreover have "\<l>\<^sub>D[F f] = F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)"
	using assms lunit_coherence by simp
	moreover have "D.iso (\<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f))"
	using assms unit_char cmp_components_are_iso
	by (intro D.isos_compose D.seqI) auto
	ultimately show ?thesis
	using assms D.invert_side_of_triangle(2) by metis
	qed
	moreover have "D.inv (\<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)) =
	(D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f))"
	using assms C.VV.arr_char unit_char FF_def D.inv_comp C.VV.dom_simp by simp
	ultimately show ?thesis by simp
	qed
	show "F \<l>\<^sub>C\<^sup>-\<^sup>1[f] = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f]"
	proof -
	have "F \<l>\<^sub>C\<^sup>-\<^sup>1[f] =
	D.inv (\<l>\<^sub>D[F f] \<cdot>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f)))"
	using assms 1 preserves_inv by simp
	also have "... = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f]"
	using assms unit_char D.comp_assoc D.isos_compose
	by (auto simp add: D.inv_comp)
	finally show ?thesis by simp
	qed
	qed

	lemma preserves_runit:
	assumes "C.ide f"
	shows "F \<r>\<^sub>C[f] = \<r>\<^sub>D[F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (unit (src\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f))"
	and "F \<r>\<^sub>C\<^sup>-\<^sup>1[f] = \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f)) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F f]"
	proof -
	show 1: "F \<r>\<^sub>C[f] = \<r>\<^sub>D[F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (unit (src\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f))"
	proof -
	have "F \<r>\<^sub>C[f] = \<r>\<^sub>D[F f] \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f)))"
	proof -
	have "\<r>\<^sub>D[F f] = F \<r>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))"
	using assms runit_coherence by simp
	moreover have "D.iso (\<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f)))"
	using assms unit_char D.iso_hcomp FF_def
	apply (intro D.isos_compose D.seqI) by auto
	moreover have "D.arr \<r>\<^sub>D[F f]"
	using assms by simp
	ultimately show ?thesis
	using assms D.invert_side_of_triangle(2) by metis
	qed
	moreover have "D.inv (\<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))) =
	(F f \<star>\<^sub>D D.inv (unit (src\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f))"
	using assms C.VV.arr_char unit_char D.iso_hcomp FF_def D.inv_comp C.VV.dom_simp
	by simp
	ultimately show ?thesis by simp
	qed
	show "F \<r>\<^sub>C\<^sup>-\<^sup>1[f] = \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f)) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F f]"
	proof -
	have "F \<r>\<^sub>C\<^sup>-\<^sup>1[f] =
	D.inv (\<r>\<^sub>D[F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (unit (src\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f)))"
	using assms 1 preserves_inv C.iso_runit by simp
	also have "... = \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f)) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F f]"
	using assms unit_char D.comp_assoc D.isos_compose
	by (auto simp add: D.inv_comp)
	finally show ?thesis by simp
	qed
	qed

	lemma preserves_assoc:
	assumes "C.ide f" and "C.ide g" and "C.ide h"
	and "src\<^sub>C f = trg\<^sub>C g" and "src\<^sub>C g = trg\<^sub>C h"
	shows "F \<a>\<^sub>C[f, g, h] = \<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h] \<cdot>\<^sub>D
	(D.inv (\<Phi> (f, g)) \<star>\<^sub>D F h) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, h))"
	and "F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] = \<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F h] \<cdot>\<^sub>D
	(F f \<star>\<^sub>D D.inv (\<Phi> (g, h))) \<cdot>\<^sub>D D.inv (\<Phi> (f, g \<star>\<^sub>C h))"
	proof -
	show 1: "F \<a>\<^sub>C[f, g, h] =
	\<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h] \<cdot>\<^sub>D
	(D.inv (\<Phi> (f, g)) \<star>\<^sub>D F h) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, h))"
	proof -
	have "F \<a>\<^sub>C[f, g, h] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h) =
	\<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h]"
	using assms assoc_coherence [of f g h] by simp
	moreover have "D.seq (\<Phi> (f, g \<star>\<^sub>C h)) ((F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h])"
	using assms C.VV.arr_char FF_def C.VV.dom_simp C.VV.cod_simp by auto
	moreover have 2: "D.iso (\<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h))"
	using assms C.VV.arr_char FF_def C.VV.dom_simp C.VV.cod_simp by auto
	moreover have "D.inv (\<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h)) =
	(D.inv (\<Phi> (f, g)) \<star>\<^sub>D F h) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, h))"
	using assms 2 C.VV.arr_char FF_def D.inv_comp C.VV.dom_simp C.VV.cod_simp
	by simp
	ultimately show ?thesis
	using D.invert_side_of_triangle(2)
	[of "\<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h]"
	"F \<a>\<^sub>C[f, g, h]" "\<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h)"]
	D.comp_assoc
	by simp
	qed
	show "F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] =
	\<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F h] \<cdot>\<^sub>D
	(F f \<star>\<^sub>D D.inv (\<Phi> (g, h))) \<cdot>\<^sub>D D.inv (\<Phi> (f, g \<star>\<^sub>C h))"
	proof -
	have "F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] =
	D.inv (\<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h] \<cdot>\<^sub>D
	(D.inv (\<Phi> (f, g)) \<star>\<^sub>D F h) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, h)))"
	using assms 1 preserves_inv by simp
	also have "... = \<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F h] \<cdot>\<^sub>D
	(F f \<star>\<^sub>D D.inv (\<Phi> (g, h))) \<cdot>\<^sub>D D.inv (\<Phi> (f, g \<star>\<^sub>C h))"
	proof -
	have "\<guillemotleft>\<Phi> (f, g \<star>\<^sub>C h) : F f \<star>\<^sub>D F (g \<star>\<^sub>C h) \<Rightarrow>\<^sub>D F (f \<star>\<^sub>C g \<star>\<^sub>C h)\<guillemotright> \<and> D.iso (\<Phi> (f, g \<star>\<^sub>C h))"
	using assms by auto
	moreover have "\<guillemotleft>F f \<star>\<^sub>D \<Phi> (g, h) : F f \<star>\<^sub>D F g \<star>\<^sub>D F h \<Rightarrow>\<^sub>D F f \<star>\<^sub>D F (g \<star>\<^sub>C h)\<guillemotright> \<and>
	D.iso (F f \<star>\<^sub>D \<Phi> (g, h))"
	using assms
	by (intro conjI D.hcomp_in_vhom, auto)
	ultimately show ?thesis
	using assms D.isos_compose D.comp_assoc
	by (elim conjE D.in_homE) (auto simp add: D.inv_comp)
	qed
	finally show ?thesis by simp
	qed
	qed

	lemma preserves_hcomp:
	assumes "C.hseq \<mu> \<nu>"
	shows "F (\<mu> \<star>\<^sub>C \<nu>) =
	\<Phi> (C.cod \<mu>, C.cod \<nu>) \<cdot>\<^sub>D (F \<mu> \<star>\<^sub>D F \<nu>) \<cdot>\<^sub>D D.inv (\<Phi> (C.dom \<mu>, C.dom \<nu>))"
	proof -
	have "F (\<mu> \<star>\<^sub>C \<nu>) \<cdot>\<^sub>D \<Phi> (C.dom \<mu>, C.dom \<nu>) = \<Phi> (C.cod \<mu>, C.cod \<nu>) \<cdot>\<^sub>D (F \<mu> \<star>\<^sub>D F \<nu>)"
	using assms \<Phi>.naturality C.VV.arr_char C.VV.cod_char FF_def C.VV.dom_simp
	by auto
	thus ?thesis
	by (metis (no_types) assms C.hcomp_simps(3) C.hseqE C.ide_dom C.src_dom C.trg_dom
	D.comp_arr_inv' D.comp_assoc cmp_components_are_iso cmp_simps(5)
	as_nat_trans.is_natural_1)
	qed

	lemma preserves_adjunction_data:
	assumes "adjunction_data_in_bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
	shows "adjunction_data_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
	(F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))
	(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
	proof
	interpret adjunction_data_in_bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>
	using assms by auto
	show "D.ide (F f)"
	using preserves_ide by simp
	show "D.ide (F g)"
	using preserves_ide by simp
	show "\<guillemotleft>D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) : src\<^sub>D (F f) \<Rightarrow>\<^sub>D F g \<star>\<^sub>D F f\<guillemotright>"
	using antipar C.VV.ide_char C.VV.arr_char cmp_in_hom(2) unit_in_hom FF_def by auto
	show "\<guillemotleft>D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) : F f \<star>\<^sub>D F g \<Rightarrow>\<^sub>D src\<^sub>D (F g)\<guillemotright>"
	using antipar C.VV.ide_char C.VV.arr_char FF_def cmp_in_hom(2) unit_in_hom(2)
	unit_char(2)
	by auto
	qed

	lemma preserves_equivalence:
	assumes "equivalence_in_bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
	shows "equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
	(F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))
	(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
	proof -
	interpret equivalence_in_bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>
	using assms by auto
	show "equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
	(F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))
	(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
	using antipar unit_is_iso preserves_iso unit_char(2) C.VV.ide_char C.VV.arr_char
	FF_def D.isos_compose
	by (unfold_locales) auto
	qed

	lemma preserves_equivalence_maps:
	assumes "C.equivalence_map f"
	shows "D.equivalence_map (F f)"
	proof -
	obtain g \<eta> \<epsilon> where E: "equivalence_in_bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
	using assms C.equivalence_map_def by auto
	interpret E: equivalence_in_bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>
	using E by auto
	show ?thesis
	using E preserves_equivalence C.equivalence_map_def D.equivalence_map_def map\<^sub>0_simps
	by blast
	qed

	lemma preserves_equivalent_objects:
	assumes "C.equivalent_objects a b"
	shows "D.equivalent_objects (map\<^sub>0 a) (map\<^sub>0 b)"
	using assms D.equivalent_objects_def preserves_equivalence_maps
	unfolding C.equivalent_objects_def by fast

	lemma preserves_isomorphic:
	assumes "C.isomorphic f g"
	shows "D.isomorphic (F f) (F g)"
	using assms C.isomorphic_def D.isomorphic_def preserves_iso by auto

	lemma preserves_quasi_inverses:
	assumes "C.quasi_inverses f g"
	shows "D.quasi_inverses (F f) (F g)"
	using assms C.quasi_inverses_def D.quasi_inverses_def preserves_equivalence by blast

	lemma preserves_quasi_inverse:
	assumes "C.equivalence_map f"
	shows "D.isomorphic (F (C.some_quasi_inverse f)) (D.some_quasi_inverse (F f))"
	using assms preserves_quasi_inverses C.quasi_inverses_some_quasi_inverse
	D.quasi_inverse_unique D.quasi_inverses_some_quasi_inverse
	preserves_equivalence_maps
	by blast

	end

	subsection "Identity Pseudofunctors"

	locale identity_pseudofunctor =
	B: bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
	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"
	begin

	text\<open>
	The underlying vertical functor is just the identity functor on the vertical category,
	which is already available as \<open>B.map\<close>.
	\<close>

	abbreviation map
	where "map \<equiv> B.map"

	interpretation I: weak_arrow_of_homs V\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>B src\<^sub>B trg\<^sub>B map
	using B.isomorphic_reflexive by unfold_locales auto

	interpretation II: "functor" B.VV.comp B.VV.comp I.FF
	using I.functor_FF by simp

	interpretation H\<^sub>BoII: composite_functor B.VV.comp B.VV.comp V\<^sub>B I.FF
	\<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>\<close>
	..
	interpretation IoH\<^sub>B: composite_functor B.VV.comp V\<^sub>B V\<^sub>B \<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>\<close> map
	..

	text\<open>
	The horizontal composition provides the compositor.
	\<close>

	abbreviation cmp
	where "cmp \<equiv> \<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>"

	interpretation cmp: natural_transformation B.VV.comp V\<^sub>B H\<^sub>BoII.map IoH\<^sub>B.map cmp
	using B.VV.arr_char B.VV.dom_simp B.VV.cod_simp B.H.as_nat_trans.is_natural_1
	B.H.as_nat_trans.is_natural_2 I.FF_def
	apply unfold_locales
	apply auto
	by (meson B.hseqE B.hseq_char')+

	interpretation cmp: natural_isomorphism B.VV.comp V\<^sub>B H\<^sub>BoII.map IoH\<^sub>B.map cmp
	by unfold_locales simp

	sublocale pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B map cmp
	apply unfold_locales
	by (metis B.assoc_is_natural_2 B.assoc_naturality B.assoc_simps(1) B.comp_ide_self
	B.hcomp_simps(1) B.ide_char B.ide_hcomp B.map_simp fst_conv snd_conv)

	lemma is_pseudofunctor:
	shows "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B map cmp"
	..

	lemma unit_char':
	assumes "B.obj a"
	shows "unit a = a"
	proof -
	have "src\<^sub>B a = a \<and> B.ide a"
	using assms by auto
	hence "a = unit a"
	using assms B.comp_arr_dom B.comp_cod_arr I.map\<^sub>0_def map_def
	B.ide_in_hom(2) B.objE [of a] B.ide_is_iso [of a]
	by (intro unit_eqI) auto
	thus ?thesis by simp
	qed

	end

	lemma (in identity_pseudofunctor) map\<^sub>0_simp [simp]:
	assumes "B.obj a"
	shows "map\<^sub>0 a = a"
	using assms map\<^sub>0_def by auto
	(* TODO: Does not recognize map\<^sub>0_def unless the context is closed, then re-opened. *)

	subsection "Embedding Pseudofunctors"

	text \<open>
	In this section, we construct the embedding pseudofunctor of a sub-bicategory
	into the ambient bicategory.
	\<close>

	locale embedding_pseudofunctor =
	B: bicategory V H \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B +
	S: subbicategory
	begin

	no_notation B.in_hom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>B _\<guillemotright>")
	notation B.in_hhom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>B _\<guillemotright>")

	definition map
	where "map \<mu> = (if S.arr \<mu> then \<mu> else B.null)"

	lemma map_in_hom [intro]:
	assumes "S.arr \<mu>"
	shows "\<guillemotleft>map \<mu> : src\<^sub>B (map (S.src \<mu>)) \<rightarrow>\<^sub>B src\<^sub>B (map (S.trg \<mu>))\<guillemotright>"
	and "\<guillemotleft>map \<mu> : map (S.dom \<mu>) \<Rightarrow>\<^sub>B map (S.cod \<mu>)\<guillemotright>"
	proof -
	show 1: "\<guillemotleft>map \<mu> : map (S.dom \<mu>) \<Rightarrow>\<^sub>B map (S.cod \<mu>)\<guillemotright>"
	using assms map_def S.in_hom_char by fastforce
	show "\<guillemotleft>map \<mu> : src\<^sub>B (map (S.src \<mu>)) \<rightarrow>\<^sub>B src\<^sub>B (map (S.trg \<mu>))\<guillemotright>"
	using assms 1 map_def S.arr_char S.src_def S.trg_def S.obj_char S.obj_src S.obj_trg
	by auto
	qed

	lemma map_simps [simp]:
	assumes "S.arr \<mu>"
	shows "B.arr (map \<mu>)"
	and "src\<^sub>B (map \<mu>) = src\<^sub>B (map (S.src \<mu>))" and "trg\<^sub>B (map \<mu>) = src\<^sub>B (map (S.trg \<mu>))"
	and "B.dom (map \<mu>) = map (S.dom \<mu>)" and "B.cod (map \<mu>) = map (S.cod \<mu>)"
	using assms map_in_hom by blast+

	interpretation "functor" S.comp V map
	apply unfold_locales
	apply auto
	using map_def S.comp_char S.seq_char S.arr_char
	apply auto[1]
	using map_def S.comp_simp by auto

	interpretation weak_arrow_of_homs S.comp S.src S.trg V src\<^sub>B trg\<^sub>B map
	using S.arr_char map_def S.src_def S.trg_def S.src_closed S.trg_closed B.isomorphic_reflexive
	preserves_ide S.ide_src S.ide_trg
	apply unfold_locales
	by presburger+

	interpretation HoFF: composite_functor S.VV.comp B.VV.comp V FF
	\<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>\<close>
	..
	interpretation FoH: composite_functor S.VV.comp S.comp V \<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star> snd \<mu>\<nu>\<close> map
	..

	no_notation B.in_hom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>B _\<guillemotright>")

	definition cmp
	where "cmp \<mu>\<nu> = (if S.VV.arr \<mu>\<nu> then fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu> else B.null)"

	lemma cmp_in_hom [intro]:
	assumes "S.VV.arr \<mu>\<nu>"
	shows "\<guillemotleft>cmp \<mu>\<nu> : src\<^sub>B (snd \<mu>\<nu>) \<rightarrow>\<^sub>B trg\<^sub>B (fst \<mu>\<nu>)\<guillemotright>"
	and "\<guillemotleft>cmp \<mu>\<nu> : map (S.dom (fst \<mu>\<nu>)) \<star>\<^sub>B map (S.dom (snd \<mu>\<nu>))
	\<Rightarrow>\<^sub>B map (S.cod (fst \<mu>\<nu>) \<star> S.cod (snd \<mu>\<nu>))\<guillemotright>"
	proof -
	show "\<guillemotleft>cmp \<mu>\<nu> : map (S.dom (fst \<mu>\<nu>)) \<star>\<^sub>B map (S.dom (snd \<mu>\<nu>))
	\<Rightarrow>\<^sub>B map (S.cod (fst \<mu>\<nu>) \<star> S.cod (snd \<mu>\<nu>))\<guillemotright>"
	proof
	show 1: "B.arr (cmp \<mu>\<nu>)"
	unfolding cmp_def
	using assms S.arr_char S.VV.arr_char S.inclusion S.src_def S.trg_def by auto
	show "B.dom (cmp \<mu>\<nu>) = map (S.dom (fst \<mu>\<nu>)) \<star>\<^sub>B map (S.dom (snd \<mu>\<nu>))"
	unfolding cmp_def map_def
	using assms 1 cmp_def S.dom_simp S.cod_simp S.VV.arr_char S.arr_char S.hcomp_def
	S.inclusion S.dom_closed
	by auto
	show "B.cod (cmp \<mu>\<nu>) = map (S.cod (fst \<mu>\<nu>) \<star> S.cod (snd \<mu>\<nu>))"
	unfolding cmp_def map_def
	using assms 1 S.H.preserves_arr S.cod_simp S.hcomp_eqI S.hcomp_simps(4) S.hseq_char'
	by auto
	qed
	thus "\<guillemotleft>cmp \<mu>\<nu> : src\<^sub>B (snd \<mu>\<nu>) \<rightarrow>\<^sub>B trg\<^sub>B (fst \<mu>\<nu>)\<guillemotright>"
	using cmp_def by auto
	qed

	lemma cmp_simps [simp]:
	assumes "S.VV.arr \<mu>\<nu>"
	shows "B.arr (cmp \<mu>\<nu>)"
	and "src\<^sub>B (cmp \<mu>\<nu>) = S.src (snd \<mu>\<nu>)" and "trg\<^sub>B (cmp \<mu>\<nu>) = S.trg (fst \<mu>\<nu>)"
	and "B.dom (cmp \<mu>\<nu>) = map (S.dom (fst \<mu>\<nu>)) \<star>\<^sub>B map (S.dom (snd \<mu>\<nu>))"
	and "B.cod (cmp \<mu>\<nu>) = map (S.cod (fst \<mu>\<nu>) \<star> S.cod (snd \<mu>\<nu>))"
	using assms cmp_in_hom S.src_def S.trg_def S.VV.arr_char
	by simp_all blast+

	lemma iso_cmp:
	assumes "S.VV.ide \<mu>\<nu>"
	shows "B.iso (cmp \<mu>\<nu>)"
	using assms S.VV.ide_char S.VV.arr_char S.arr_char cmp_def S.ide_char S.src_def S.trg_def
	by auto

	interpretation \<Phi>\<^sub>E: natural_isomorphism S.VV.comp V HoFF.map FoH.map cmp
	proof
	show "\<And>\<mu>\<nu>. \<not> S.VV.arr \<mu>\<nu> \<Longrightarrow> cmp \<mu>\<nu> = B.null"
	using cmp_def by simp
	fix \<mu>\<nu>
	assume \<mu>\<nu>: "S.VV.arr \<mu>\<nu>"
	have 1: "S.arr (fst \<mu>\<nu>) \<and> S.arr (snd \<mu>\<nu>) \<and> S.src (fst \<mu>\<nu>) = S.trg (snd \<mu>\<nu>)"
	using \<mu>\<nu> S.VV.arr_char by simp
	show "B.dom (cmp \<mu>\<nu>) = HoFF.map (S.VV.dom \<mu>\<nu>)"
	using \<mu>\<nu> FF_def S.VV.arr_char S.VV.dom_char S.arr_dom S.src_def S.trg_def
	S.dom_char S.src.preserves_dom S.trg.preserves_dom
	apply simp
	by (metis (no_types, lifting))
	show "B.cod (cmp \<mu>\<nu>) = FoH.map (S.VV.cod \<mu>\<nu>)"
	using \<mu>\<nu> 1 map_def S.hseq_char S.hcomp_def S.cod_char S.arr_cod S.VV.cod_simp
	by simp
	show "cmp (S.VV.cod \<mu>\<nu>) \<cdot>\<^sub>B HoFF.map \<mu>\<nu> = cmp \<mu>\<nu>"
	using \<mu>\<nu> 1 cmp_def S.VV.arr_char S.VV.cod_char FF_def S.arr_cod S.cod_simp
	S.src_def S.trg_def map_def
	apply simp
	by (metis (no_types, lifting) B.comp_cod_arr B.hcomp_simps(4) cmp_simps(1) \<mu>\<nu>)
	show "FoH.map \<mu>\<nu> \<cdot>\<^sub>B cmp (S.VV.dom \<mu>\<nu>) = cmp \<mu>\<nu>"
	unfolding cmp_def map_def
	using \<mu>\<nu> S.VV.arr_char B.VV.arr_char S.VV.dom_char S.VV.cod_char B.comp_arr_dom
	S.hcomp_def
	apply simp
	by (metis (no_types, lifting) B.hcomp_simps(3) cmp_def cmp_simps(1) S.arr_char
	S.dom_char S.hcomp_closed S.src_def S.trg_def)
	next
	show "\<And>fg. S.VV.ide fg \<Longrightarrow> B.iso (cmp fg)"
	using iso_cmp by simp
	qed

	sublocale pseudofunctor S.comp S.hcomp S.\<a> \<i> S.src S.trg V H \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B map cmp
	proof
	fix f g h
	assume f: "S.ide f" and g: "S.ide g" and h: "S.ide h"
	and fg: "S.src f = S.trg g" and gh: "S.src g = S.trg h"
	have 1: "B.ide f \<and> B.ide g \<and> B.ide h \<and> src\<^sub>B f = trg\<^sub>B g \<and> src\<^sub>B g = trg\<^sub>B h"
	using f g h fg gh S.ide_char S.arr_char S.src_def S.trg_def by simp
	show "map (S.\<a> f g h) \<cdot>\<^sub>B cmp (f \<star> g, h) \<cdot>\<^sub>B cmp (f, g) \<star>\<^sub>B map h =
	cmp (f, g \<star> h) \<cdot>\<^sub>B (map f \<star>\<^sub>B cmp (g, h)) \<cdot>\<^sub>B \<a>\<^sub>B[map f, map g, map h]"
	proof -
	have "map (S.\<a> f g h) \<cdot>\<^sub>B cmp (f \<star> g, h) \<cdot>\<^sub>B cmp (f, g) \<star>\<^sub>B map h =
	\<a>\<^sub>B[f, g, h] \<cdot>\<^sub>B ((f \<star>\<^sub>B g) \<star>\<^sub>B h) \<cdot>\<^sub>B ((f \<star>\<^sub>B g) \<star>\<^sub>B h)"
	unfolding map_def cmp_def
	using 1 f g h fg gh S.VVV.arr_char S.VV.arr_char B.VVV.arr_char B.VV.arr_char
	B.comp_arr_dom S.arr_char S.comp_char S.hcomp_closed S.hcomp_def
	S.ideD(1) S.src_def
	by (simp add: S.assoc_closed)
	also have "... = cmp (f, g \<star> h) \<cdot>\<^sub>B (map f \<star>\<^sub>B cmp (g, h)) \<cdot>\<^sub>B \<a>\<^sub>B[map f, map g, map h]"
	unfolding cmp_def map_def
	using 1 f g h fg gh S.VV.arr_char B.VVV.arr_char B.VV.arr_char
	B.comp_arr_dom B.comp_cod_arr S.hcomp_def S.comp_char
	S.arr_char S.assoc_closed S.hcomp_closed S.ideD(1) S.trg_def
	by auto
	finally show ?thesis by blast
	qed
	qed

	lemma is_pseudofunctor:
	shows "pseudofunctor S.comp S.hcomp S.\<a> \<i> S.src S.trg V H \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B map cmp"
	..

	lemma map\<^sub>0_simp [simp]:
	assumes "S.obj a"
	shows "map\<^sub>0 a = a"
	using assms map\<^sub>0_def map_def S.obj_char by auto

	lemma unit_char':
	assumes "S.obj a"
	shows "unit a = a"
	proof -
	have a: "S.arr a"
	using assms by auto
	have 1: "B.ide a"
	using assms S.obj_char by auto
	have 2: "src\<^sub>B a = a"
	using assms S.obj_char by auto
	have "a = unit a"
	proof (intro unit_eqI)
	show "S.obj a" by fact
	show "\<guillemotleft>a : map\<^sub>0 a \<Rightarrow>\<^sub>B map a\<guillemotright>"
	using assms a 2 map\<^sub>0_def map_def S.src_def S.dom_char S.cod_char S.obj_char
	by auto
	show "B.iso a"
	using assms 1 by simp
	show "a \<cdot>\<^sub>B \<i>[map\<^sub>0 a] = (map \<i>[a] \<cdot>\<^sub>B cmp (a, a)) \<cdot>\<^sub>B (a \<star>\<^sub>B a)"
	proof -
	have "a \<cdot>\<^sub>B \<i>[a] = \<i>[a] \<cdot>\<^sub>B cmp (a, a) \<cdot>\<^sub>B (a \<star>\<^sub>B a)"
	proof -
	have "a \<cdot>\<^sub>B \<i>[a] = \<i>[a]"
	using assms 1 2 S.comp_cod_arr S.unitor_coincidence S.lunit_in_hom
	S.obj_char S.comp_simp
	by auto
	moreover have "(a \<star>\<^sub>B a) \<cdot>\<^sub>B (a \<star>\<^sub>B a) = a \<star>\<^sub>B a"
	using assms S.obj_char S.comp_ide_self by auto
	moreover have "B.dom \<i>[a] = a \<star>\<^sub>B a"
	using assms S.obj_char by simp
	moreover have "\<i>[a] \<cdot>\<^sub>B (a \<star>\<^sub>B a) = \<i>[a]"
	using assms S.obj_char B.comp_arr_dom by simp
	ultimately show ?thesis
	using assms cmp_def S.VV.arr_char by auto
	qed
	thus ?thesis
	using assms a 2 map\<^sub>0_def map_def S.src_def B.comp_assoc by simp
	qed
	qed
	thus ?thesis by simp
	qed

	end

	subsection "Composition of Pseudofunctors"

	text \<open>
	In this section, we show how pseudofunctors may be composed. The main work is to
	establish the coherence condition for associativity.
	\<close>

	locale composite_pseudofunctor =
	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 +
	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 +
	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>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 :: "'c \<Rightarrow> 'd"
	and \<Phi>\<^sub>G :: "'c * 'c \<Rightarrow> 'd"
	begin

	sublocale composite_functor V\<^sub>B V\<^sub>C V\<^sub>D F G ..

	sublocale weak_arrow_of_homs V\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D src\<^sub>D trg\<^sub>D \<open>G o F\<close>
	proof
	show "\<And>\<mu>. B.arr \<mu> \<Longrightarrow> D.isomorphic ((G o F) (src\<^sub>B \<mu>)) (src\<^sub>D ((G o F) \<mu>))"
	proof -
	fix \<mu>
	assume \<mu>: "B.arr \<mu>"
	show "D.isomorphic ((G o F) (src\<^sub>B \<mu>)) (src\<^sub>D ((G o F) \<mu>))"
	proof -
	have "(G o F) (src\<^sub>B \<mu>) = G (F (src\<^sub>B \<mu>))"
	using \<mu> by simp
	also have "D.isomorphic ... (G (src\<^sub>C (F \<mu>)))"
	using \<mu> F.weakly_preserves_src G.preserves_iso
	by (meson C.isomorphicE D.isomorphic_def G.preserves_hom)
	also have "D.isomorphic ... (src\<^sub>D (G (F \<mu>)))"
	using \<mu> G.weakly_preserves_src by blast
	also have "... = src\<^sub>D ((G o F) \<mu>)"
	by simp
	finally show ?thesis by blast
	qed
	qed
	show "\<And>\<mu>. B.arr \<mu> \<Longrightarrow> D.isomorphic ((G o F) (trg\<^sub>B \<mu>)) (trg\<^sub>D ((G o F) \<mu>))"
	proof -
	fix \<mu>
	assume \<mu>: "B.arr \<mu>"
	show "D.isomorphic ((G o F) (trg\<^sub>B \<mu>)) (trg\<^sub>D ((G o F) \<mu>))"
	proof -
	have "(G o F) (trg\<^sub>B \<mu>) = G (F (trg\<^sub>B \<mu>))"
	using \<mu> by simp
	also have "D.isomorphic ... (G (trg\<^sub>C (F \<mu>)))"
	using \<mu> F.weakly_preserves_trg G.preserves_iso
	by (meson C.isomorphicE D.isomorphic_def G.preserves_hom)
	also have "D.isomorphic ... (trg\<^sub>D (G (F \<mu>)))"
	using \<mu> G.weakly_preserves_trg by blast
	also have "... = trg\<^sub>D ((G o F) \<mu>)"
	by simp
	finally show ?thesis by blast
	qed
	qed
	qed

	interpretation H\<^sub>DoGF_GF: composite_functor B.VV.comp D.VV.comp V\<^sub>D FF
	\<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>D snd \<mu>\<nu>\<close>
	..
	interpretation GFoH\<^sub>B: composite_functor B.VV.comp V\<^sub>B V\<^sub>D \<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>\<close>
	\<open>G o F\<close>
	..

	definition cmp
	where "cmp \<mu>\<nu> = (if B.VV.arr \<mu>\<nu> then
	G (F (H\<^sub>B (fst \<mu>\<nu>) (snd \<mu>\<nu>))) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (B.dom (fst \<mu>\<nu>)), F (B.dom (snd \<mu>\<nu>)))
	else D.null)"

	lemma cmp_in_hom [intro]:
	assumes "B.VV.arr \<mu>\<nu>"
	shows "\<guillemotleft>cmp \<mu>\<nu> : H\<^sub>DoGF_GF.map (B.VV.dom \<mu>\<nu>) \<Rightarrow>\<^sub>D GFoH\<^sub>B.map (B.VV.cod \<mu>\<nu>)\<guillemotright>"
	proof -
	have "cmp \<mu>\<nu> = G (F (H\<^sub>B (fst \<mu>\<nu>) (snd \<mu>\<nu>))) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (B.dom (fst \<mu>\<nu>)), F (B.dom (snd \<mu>\<nu>)))"
	using assms cmp_def by simp
	moreover have "\<guillemotleft> ... : H\<^sub>DoGF_GF.map (B.VV.dom \<mu>\<nu>)
	\<Rightarrow>\<^sub>D GFoH\<^sub>B.map (B.VV.cod \<mu>\<nu>)\<guillemotright>"
	proof (intro D.comp_in_homI)
	show "\<guillemotleft>\<Phi>\<^sub>G (F (B.dom (fst \<mu>\<nu>)), F (B.dom (snd \<mu>\<nu>))) :
	H\<^sub>DoGF_GF.map (B.VV.dom \<mu>\<nu>)
	\<Rightarrow>\<^sub>D G (F (B.dom (fst \<mu>\<nu>)) \<star>\<^sub>C F (B.dom (snd \<mu>\<nu>)))\<guillemotright>"
	using assms F.FF_def FF_def B.VV.arr_char B.VV.dom_simp by auto
	show "\<guillemotleft>G (\<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>)) :
	G (F (B.dom (fst \<mu>\<nu>)) \<star>\<^sub>C F (B.dom (snd \<mu>\<nu>)))
	\<Rightarrow>\<^sub>D GFoH\<^sub>B.map (B.VV.dom \<mu>\<nu>)\<guillemotright>"
	using assms B.VV.arr_char F.FF_def B.VV.dom_simp B.VV.cod_simp by auto
	show "\<guillemotleft>G (F (fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>)) :
	GFoH\<^sub>B.map (B.VV.dom \<mu>\<nu>) \<Rightarrow>\<^sub>D GFoH\<^sub>B.map (B.VV.cod \<mu>\<nu>)\<guillemotright>"
	proof -
	have "B.VV.in_hom \<mu>\<nu> (B.VV.dom \<mu>\<nu>) (B.VV.cod \<mu>\<nu>)"
	using assms by auto
	thus ?thesis by auto
	qed
	qed
	ultimately show ?thesis by argo
	qed

	lemma cmp_simps [simp]:
	assumes "B.VV.arr \<mu>\<nu>"
	shows "D.arr (cmp \<mu>\<nu>)"
	and "D.dom (cmp \<mu>\<nu>) = H\<^sub>DoGF_GF.map (B.VV.dom \<mu>\<nu>)"
	and "D.cod (cmp \<mu>\<nu>) = GFoH\<^sub>B.map (B.VV.cod \<mu>\<nu>)"
	using assms cmp_in_hom by blast+

	interpretation \<Phi>: natural_transformation
	B.VV.comp V\<^sub>D H\<^sub>DoGF_GF.map GFoH\<^sub>B.map cmp
	proof
	show "\<And>\<mu>\<nu>. \<not> B.VV.arr \<mu>\<nu> \<Longrightarrow> cmp \<mu>\<nu> = D.null"
	unfolding cmp_def by simp
	fix \<mu>\<nu>
	assume \<mu>\<nu>: "B.VV.arr \<mu>\<nu>"
	show "D.dom (cmp \<mu>\<nu>) = H\<^sub>DoGF_GF.map (B.VV.dom \<mu>\<nu>)"
	using \<mu>\<nu> cmp_in_hom by blast
	show "D.cod (cmp \<mu>\<nu>) = GFoH\<^sub>B.map (B.VV.cod \<mu>\<nu>)"
	using \<mu>\<nu> cmp_in_hom by blast
	show "GFoH\<^sub>B.map \<mu>\<nu> \<cdot>\<^sub>D cmp (B.VV.dom \<mu>\<nu>) = cmp \<mu>\<nu>"
	unfolding cmp_def
	using \<mu>\<nu> B.VV.ide_char B.VV.arr_char D.comp_ide_arr B.VV.dom_char D.comp_assoc
	as_nat_trans.is_natural_1
	apply simp
	by (metis (no_types, lifting) B.H.preserves_arr B.hcomp_simps(3))
	show "cmp (B.VV.cod \<mu>\<nu>) \<cdot>\<^sub>D H\<^sub>DoGF_GF.map \<mu>\<nu> = cmp \<mu>\<nu>"
	proof -
	have "cmp (B.VV.cod \<mu>\<nu>) \<cdot>\<^sub>D H\<^sub>DoGF_GF.map \<mu>\<nu> =
	(G (F (B.cod (fst \<mu>\<nu>) \<star>\<^sub>B B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (B.cod (fst \<mu>\<nu>)), F (B.cod (snd \<mu>\<nu>)))) \<cdot>\<^sub>D
	(fst (FF \<mu>\<nu>) \<star>\<^sub>D snd (FF \<mu>\<nu>))"
	unfolding cmp_def
	using \<mu>\<nu> B.VV.arr_char B.VV.dom_simp B.VV.cod_simp by simp
	also have "... = (G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (B.cod (fst \<mu>\<nu>)), F (B.cod (snd \<mu>\<nu>)))) \<cdot>\<^sub>D
	(fst (FF \<mu>\<nu>) \<star>\<^sub>D snd (FF \<mu>\<nu>))"
	proof -
	have "D.ide (G (F (B.cod (fst \<mu>\<nu>) \<star>\<^sub>B B.cod (snd \<mu>\<nu>))))"
	using \<mu>\<nu> B.VV.arr_char by simp
	moreover have "D.seq (G (F (B.cod (fst \<mu>\<nu>) \<star>\<^sub>B B.cod (snd \<mu>\<nu>))))
	(G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (B.cod (fst \<mu>\<nu>)), F (B.cod (snd \<mu>\<nu>))))"
	using \<mu>\<nu> B.VV.arr_char B.VV.dom_char B.VV.cod_char F.FF_def
	apply (intro D.seqI)
	apply auto
	proof -
	show "G (F (B.cod (fst \<mu>\<nu>) \<star>\<^sub>B B.cod (snd \<mu>\<nu>))) =
	D.cod (G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (B.cod (fst \<mu>\<nu>)), F (B.cod (snd \<mu>\<nu>))))"
	proof -
	have "D.seq (G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))))
	(\<Phi>\<^sub>G (F (B.cod (fst \<mu>\<nu>)), F (B.cod (snd \<mu>\<nu>))))"
	using \<mu>\<nu> B.VV.arr_char F.FF_def B.VV.arr_char B.VV.dom_char B.VV.cod_char
	by (intro D.seqI) auto
	thus ?thesis
	using \<mu>\<nu> B.VV.arr_char B.VV.cod_simp by simp
	qed
	qed
	ultimately show ?thesis
	using \<mu>\<nu> D.comp_ide_arr [of "G (F (B.cod (fst \<mu>\<nu>) \<star>\<^sub>B B.cod (snd \<mu>\<nu>)))"
	"G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (B.cod (fst \<mu>\<nu>)), F (B.cod (snd \<mu>\<nu>)))"]
	by simp
	qed
	also have "... = G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	(\<Phi>\<^sub>G (F (B.cod (fst \<mu>\<nu>)), F (B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	(fst (FF \<mu>\<nu>) \<star>\<^sub>D snd (FF \<mu>\<nu>)))"
	using D.comp_assoc by simp
	also have "... = G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (C.VV.cod (F.FF \<mu>\<nu>)) \<cdot>\<^sub>D G.H\<^sub>DoFF.map (F.FF \<mu>\<nu>)"
	using \<mu>\<nu> B.VV.arr_char F.FF_def G.FF_def FF_def C.VV.cod_simp by auto
	also have "... = G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	G.FoH\<^sub>C.map (F.FF \<mu>\<nu>) \<cdot>\<^sub>D \<Phi>\<^sub>G (C.VV.dom (F.FF \<mu>\<nu>))"
	using \<mu>\<nu> B.VV.arr_char G.\<Phi>.naturality C.VV.dom_simp C.VV.cod_simp by simp
	also have "... = (G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D
	G.FoH\<^sub>C.map (F.FF \<mu>\<nu>)) \<cdot>\<^sub>D \<Phi>\<^sub>G (C.VV.dom (F.FF \<mu>\<nu>))"
	using D.comp_assoc by simp
	also have "... = (G (\<Phi>\<^sub>F (B.VV.cod \<mu>\<nu>) \<cdot>\<^sub>C F.H\<^sub>DoFF.map \<mu>\<nu>)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (C.VV.dom (F.FF \<mu>\<nu>))"
	proof -
	have "(B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>)) = B.VV.cod \<mu>\<nu>"
	using \<mu>\<nu> B.VV.arr_char B.VV.cod_simp by simp
	moreover have "G.FoH\<^sub>C.map (F.FF \<mu>\<nu>) = G (F.H\<^sub>DoFF.map \<mu>\<nu>)"
	using \<mu>\<nu> F.FF_def by simp
	moreover have "G (\<Phi>\<^sub>F (B.cod (fst \<mu>\<nu>), B.cod (snd \<mu>\<nu>))) \<cdot>\<^sub>D G (F.H\<^sub>DoFF.map \<mu>\<nu>) =
	G (\<Phi>\<^sub>F (B.VV.cod \<mu>\<nu>) \<cdot>\<^sub>C F.H\<^sub>DoFF.map \<mu>\<nu>)"
	using \<mu>\<nu> B.VV.arr_char
	by (metis (no_types, lifting) F.\<Phi>.is_natural_2 F.\<Phi>.preserves_reflects_arr
	G.preserves_comp calculation(1))
	ultimately show ?thesis by argo
	qed
	also have "... = G (F.FoH\<^sub>C.map \<mu>\<nu> \<cdot>\<^sub>C \<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (C.VV.dom (F.FF \<mu>\<nu>))"
	using \<mu>\<nu> F.\<Phi>.naturality [of \<mu>\<nu>] F.FF_def by simp
	also have "... = G (F.FoH\<^sub>C.map \<mu>\<nu>) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (C.VV.dom (F.FF \<mu>\<nu>))"
	proof -
	have "G (F.FoH\<^sub>C.map \<mu>\<nu> \<cdot>\<^sub>C \<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>)) =
	G (F.FoH\<^sub>C.map \<mu>\<nu>) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>))"
	using \<mu>\<nu>
	by (metis (mono_tags, lifting) F.\<Phi>.is_natural_1 F.\<Phi>.preserves_reflects_arr
	G.preserves_comp)
	thus ?thesis
	using \<mu>\<nu> D.comp_assoc by simp
	qed
	also have "... = cmp \<mu>\<nu>"
	using \<mu>\<nu> B.VV.arr_char cmp_def F.FF_def F.FF.preserves_dom B.VV.dom_simp
	by auto
	finally show ?thesis by simp
	qed
	qed

	interpretation \<Phi>: natural_isomorphism B.VV.comp V\<^sub>D H\<^sub>DoGF_GF.map GFoH\<^sub>B.map cmp
	proof
	show "\<And>hk. B.VV.ide hk \<Longrightarrow> D.iso (cmp hk)"
	proof -
	fix hk
	assume hk: "B.VV.ide hk"
	have "D.iso (G (F (fst hk \<star>\<^sub>B snd hk)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (B.VV.dom hk)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (B.dom (fst hk)), F (B.dom (snd hk))))"
	proof (intro D.isos_compose)
	show "D.iso (\<Phi>\<^sub>G (F (B.dom (fst hk)), F (B.dom (snd hk))))"
	using hk G.\<Phi>.components_are_iso [of "(F (B.dom (fst hk)), F (B.dom (snd hk)))"]
	C.VV.ide_char B.VV.arr_char B.VV.dom_char
	by (metis (no_types, lifting) B.VV.ideD(1) B.VV.ideD(2) B.VxV.dom_char
	F.FF_def F.FF.as_nat_iso.components_are_iso G.\<Phi>.preserves_iso fst_conv snd_conv)
	show "D.iso (G (\<Phi>\<^sub>F (B.VV.dom hk)))"
	using hk F.\<Phi>.components_are_iso B.VV.arr_char B.VV.dom_char B.VV.ideD(2)
	by auto
	show "D.seq (G (\<Phi>\<^sub>F (B.VV.dom hk))) (\<Phi>\<^sub>G (F (B.dom (fst hk)), F (B.dom (snd hk))))"
	using hk B.VV.arr_char B.VV.ide_char B.VV.dom_char C.VV.arr_char F.FF_def
	C.VV.dom_simp C.VV.cod_simp
	by auto
	show "D.iso (G (F (fst hk \<star>\<^sub>B snd hk)))"
	using hk F.\<Phi>.components_are_iso B.VV.arr_char by simp
	show "D.seq (G (F (fst hk \<star>\<^sub>B snd hk)))
	(G (\<Phi>\<^sub>F (B.VV.dom hk)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F (B.dom (fst hk)), F (B.dom (snd hk))))"
	using hk B.VV.arr_char B.VV.dom_char
	by (metis (no_types, lifting) B.VV.ideD(1) cmp_def cmp_simps(1))
	qed
	thus "D.iso (cmp hk)"
	unfolding cmp_def using hk by simp
	qed
	qed

	sublocale pseudofunctor 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>G o F\<close> cmp
	proof
	fix f g h
	assume f: "B.ide f" and g: "B.ide g" and h: "B.ide h"
	assume fg: "src\<^sub>B f = trg\<^sub>B g" and gh: "src\<^sub>B g = trg\<^sub>B h"
	show "map \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>D cmp (f \<star>\<^sub>B g, h) \<cdot>\<^sub>D (cmp (f, g) \<star>\<^sub>D map h) =
	cmp (f, g \<star>\<^sub>B h) \<cdot>\<^sub>D (map f \<star>\<^sub>D cmp (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[map f, map g, map h]"
	proof -
	have "map \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>D cmp (f \<star>\<^sub>B g, h) \<cdot>\<^sub>D (cmp (f, g) \<star>\<^sub>D map h) =
	G (F \<a>\<^sub>B[f, g, h]) \<cdot>\<^sub>D
	(G (F ((f \<star>\<^sub>B g) \<star>\<^sub>B h)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F (f \<star>\<^sub>B g), F h)) \<cdot>\<^sub>D
	(G (F (f \<star>\<^sub>B g)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (f, g)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	unfolding cmp_def
	using f g h fg gh B.VV.arr_char B.VV.dom_simp by simp
	also have "... = G (F \<a>\<^sub>B[f, g, h]) \<cdot>\<^sub>D
	(G (\<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F (f \<star>\<^sub>B g), F h)) \<cdot>\<^sub>D
	(G (F (f \<star>\<^sub>B g)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (f, g)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using f g h fg gh D.comp_ide_arr D.comp_assoc
	by (metis B.ideD(1) B.ide_hcomp B.src_hcomp F.cmp_simps(1) F.cmp_simps(5)
	G.as_nat_trans.is_natural_2)
	also have "... = G (F \<a>\<^sub>B[f, g, h]) \<cdot>\<^sub>D
	(G (\<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F (f \<star>\<^sub>B g), F h)) \<cdot>\<^sub>D
	(G (\<Phi>\<^sub>F (f, g)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using f g fg
	by (metis (no_types) D.comp_assoc F.cmp_simps(1) F.cmp_simps(5)
	G.as_nat_trans.is_natural_2)
	also have "... = (G (F \<a>\<^sub>B[f, g, h]) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (f \<star>\<^sub>B g, h))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (f \<star>\<^sub>B g), F h) \<cdot>\<^sub>D (G (\<Phi>\<^sub>F (f, g)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using D.comp_assoc by simp
	also have "... = G (F \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>C \<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (f \<star>\<^sub>B g), F h) \<cdot>\<^sub>D (G (\<Phi>\<^sub>F (f, g)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using f g h fg gh B.VV.arr_char B.VV.cod_simp by simp
	also have "... = G (F \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>C \<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F (f \<star>\<^sub>B g), F h) \<cdot>\<^sub>D (G (\<Phi>\<^sub>F (f, g)) \<star>\<^sub>D G (F h)) \<cdot>\<^sub>D
	(\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using f g h fg gh B.VV.arr_char C.VV.arr_char F.FF_def D.whisker_right
	B.VV.dom_simp C.VV.cod_simp
	by auto
	also have "... = G (F \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>C \<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>D
	(\<Phi>\<^sub>G (F (f \<star>\<^sub>B g), F h) \<cdot>\<^sub>D (G (\<Phi>\<^sub>F (f, g)) \<star>\<^sub>D G (F h))) \<cdot>\<^sub>D
	(\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using D.comp_assoc by simp
	also have "... = G (F \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>C \<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>D
	(G (\<Phi>\<^sub>F (f, g) \<star>\<^sub>C F h) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h)) \<cdot>\<^sub>D
	(\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	proof -
	have "\<Phi>\<^sub>G (F (f \<star>\<^sub>B g), F h) = \<Phi>\<^sub>G (C.VV.cod (\<Phi>\<^sub>F (f, g), F h))"
	using f g h fg gh B.VV.arr_char C.VV.arr_char B.VV.cod_simp C.VV.cod_simp
	by simp
	moreover have "G (\<Phi>\<^sub>F (f, g)) \<star>\<^sub>D G (F h) = G.H\<^sub>DoFF.map (\<Phi>\<^sub>F (f, g), F h)"
	using f g h fg gh B.VV.arr_char C.VV.arr_char G.FF_def by simp
	moreover have "G.FoH\<^sub>C.map (\<Phi>\<^sub>F (f, g), F h) = G (\<Phi>\<^sub>F (f, g) \<star>\<^sub>C F h)"
	using f g h fg gh B.VV.arr_char by simp
	moreover have "\<Phi>\<^sub>G (C.VV.dom (\<Phi>\<^sub>F (f, g), F h)) = \<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h)"
	using f g h fg gh C.VV.arr_char F.cmp_in_hom [of f g] C.VV.dom_simp by auto
	ultimately show ?thesis
	using f g h fg gh B.VV.arr_char G.\<Phi>.naturality
	by (metis (mono_tags, lifting) C.VV.arr_cod_iff_arr C.VV.arr_dom_iff_arr
	G.FoH\<^sub>C.is_extensional G.H\<^sub>DoFF.is_extensional G.\<Phi>.is_extensional)
	qed
	also have "... = (G (F \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>C \<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>D (G (\<Phi>\<^sub>F (f, g) \<star>\<^sub>C F h))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h) \<cdot>\<^sub>D (\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using D.comp_assoc by simp
	also have "... = G ((F \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>C \<Phi>\<^sub>F (f \<star>\<^sub>B g, h)) \<cdot>\<^sub>C (\<Phi>\<^sub>F (f, g) \<star>\<^sub>C F h)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h) \<cdot>\<^sub>D (\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using f g h fg gh B.VV.arr_char F.FF_def B.VV.dom_simp B.VV.cod_simp by auto
	also have "... = G (F \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>C \<Phi>\<^sub>F (f \<star>\<^sub>B g, h) \<cdot>\<^sub>C (\<Phi>\<^sub>F (f, g) \<star>\<^sub>C F h)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h) \<cdot>\<^sub>D (\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using C.comp_assoc by simp
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h)) \<cdot>\<^sub>C \<a>\<^sub>C[F f, F g, F h]) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h) \<cdot>\<^sub>D (\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using f g h fg gh F.assoc_coherence [of f g h] by simp
	also have "... = G ((\<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h))) \<cdot>\<^sub>C \<a>\<^sub>C[F f, F g, F h]) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h) \<cdot>\<^sub>D (\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using C.comp_assoc by simp
	also have "... = (G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h))) \<cdot>\<^sub>D G \<a>\<^sub>C[F f, F g, F h]) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h) \<cdot>\<^sub>D (\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using f g h fg gh B.VV.arr_char F.FF_def B.VV.dom_simp B.VV.cod_simp by auto
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h))) \<cdot>\<^sub>D
	G \<a>\<^sub>C[F f, F g, F h] \<cdot>\<^sub>D \<Phi>\<^sub>G (F f \<star>\<^sub>C F g, F h) \<cdot>\<^sub>D
	(\<Phi>\<^sub>G (F f, F g) \<star>\<^sub>D G (F h))"
	using D.comp_assoc by simp
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f, F g \<star>\<^sub>C F h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D \<Phi>\<^sub>G (F g, F h)) \<cdot>\<^sub>D
	\<a>\<^sub>D[G (F f), G (F g), G (F h)]"
	using f g h fg gh G.assoc_coherence [of "F f" "F g" "F h"] by simp
	also have "... = (G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f, F g \<star>\<^sub>C F h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D \<Phi>\<^sub>G (F g, F h))) \<cdot>\<^sub>D
	\<a>\<^sub>D[G (F f), G (F g), G (F h)]"
	using D.comp_assoc by simp
	also have "... = (cmp (f, g \<star>\<^sub>B h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D cmp (g, h))) \<cdot>\<^sub>D
	\<a>\<^sub>D[G (F f), G (F g), G (F h)]"
	proof -
	have "G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f, F g \<star>\<^sub>C F h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D \<Phi>\<^sub>G (F g, F h)) =
	cmp (f, g \<star>\<^sub>B h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D cmp (g, h))"
	proof -
	have "cmp (f, g \<star>\<^sub>B h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D cmp (g, h)) =
	(G (F (f \<star>\<^sub>B g \<star>\<^sub>B h)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F (g \<star>\<^sub>B h))) \<cdot>\<^sub>D
	(G (F f) \<star>\<^sub>D G (F (g \<star>\<^sub>B h)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (g, h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	unfolding cmp_def
	using f g h fg gh B.VV.arr_char B.VV.dom_simp B.VV.cod_simp by simp
	also have "... = (G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F (g \<star>\<^sub>B h))) \<cdot>\<^sub>D
	(G (F f) \<star>\<^sub>D G (F (g \<star>\<^sub>B h)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (g, h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	proof -
	have "G (F (f \<star>\<^sub>B g \<star>\<^sub>B h)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h))"
	using f g h fg gh B.VV.arr_char D.comp_ide_arr B.VV.dom_simp B.VV.cod_simp
	by simp
	thus ?thesis
	using D.comp_assoc by metis
	qed
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F (g \<star>\<^sub>B h)) \<cdot>\<^sub>D
	(G (F f) \<star>\<^sub>D G (F (g \<star>\<^sub>B h)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (g, h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	using D.comp_assoc by simp
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F (g \<star>\<^sub>B h)) \<cdot>\<^sub>D
	(G (F f) \<star>\<^sub>D G (\<Phi>\<^sub>F (g, h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	proof -
	have "G (F (g \<star>\<^sub>B h)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (g, h)) = G (\<Phi>\<^sub>F (g, h))"
	using f g h fg gh B.VV.arr_char D.comp_ide_arr B.VV.dom_simp B.VV.cod_simp
	by simp
	thus ?thesis
	using D.comp_assoc by metis
	qed
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F (g \<star>\<^sub>B h)) \<cdot>\<^sub>D
	(G (F f) \<star>\<^sub>D G (\<Phi>\<^sub>F (g, h))) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	using f g h fg gh
	D.whisker_left [of "G (F f)" "G (\<Phi>\<^sub>F (g, h))" "\<Phi>\<^sub>G (F g, F h)"]
	by fastforce
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) \<cdot>\<^sub>D
	(\<Phi>\<^sub>G (F f, F (g \<star>\<^sub>B h)) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D G (\<Phi>\<^sub>F (g, h)))) \<cdot>\<^sub>D
	(G (F f) \<star>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	using D.comp_assoc by simp
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) \<cdot>\<^sub>D
	(G (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F f, F g \<star>\<^sub>C F h)) \<cdot>\<^sub>D
	(G (F f) \<star>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	proof -
	have "\<Phi>\<^sub>G (C.VV.cod (F f, \<Phi>\<^sub>F (g, h))) = \<Phi>\<^sub>G (F f, F (g \<star>\<^sub>B h))"
	using f g h fg gh B.VV.arr_char C.VV.cod_char B.VV.dom_simp B.VV.cod_simp
	by auto
	moreover have "G.H\<^sub>DoFF.map (F f, \<Phi>\<^sub>F (g, h)) = G (F f) \<star>\<^sub>D G (\<Phi>\<^sub>F (g, h))"
	using f g h fg gh B.VV.arr_char G.FF_def by auto
	moreover have "G.FoH\<^sub>C.map (F f, \<Phi>\<^sub>F (g, h)) = G (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h))"
	using f g h fg gh B.VV.arr_char C.VV.arr_char by simp
	moreover have "C.VV.dom (F f, \<Phi>\<^sub>F (g, h)) = (F f, F g \<star>\<^sub>C F h)"
	using f g h fg gh B.VV.arr_char C.VV.arr_char C.VV.dom_char
	F.cmp_in_hom [of g h]
	by auto
	ultimately show ?thesis
	using f g h fg gh B.VV.arr_char C.VV.arr_char
	G.\<Phi>.naturality [of "(F f, \<Phi>\<^sub>F (g, h))"]
	by simp
	qed
	also have "... = (G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h)) \<cdot>\<^sub>D (G (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h)))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f, F g \<star>\<^sub>C F h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	using D.comp_assoc by simp
	also have "... = G (\<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F f, F g \<star>\<^sub>C F h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D \<Phi>\<^sub>G (F g, F h))"
	using f g h fg gh B.VV.arr_char
	by (metis (no_types, lifting) B.assoc_simps(1) C.comp_assoc C.seqE
	F.preserves_assoc(1) F.preserves_reflects_arr G.preserves_comp)
	finally show ?thesis by simp
	qed
	thus ?thesis by simp
	qed
	also have "... = cmp (f, g \<star>\<^sub>B h) \<cdot>\<^sub>D (G (F f) \<star>\<^sub>D cmp (g, h)) \<cdot>\<^sub>D
	\<a>\<^sub>D[G (F f), G (F g), G (F h)]"
	using D.comp_assoc by simp
	finally show ?thesis by simp
	qed
	qed

	lemma is_pseudofunctor:
	shows "pseudofunctor 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 (G o F) cmp"
	..

	lemma map\<^sub>0_simp [simp]:
	assumes "B.obj a"
	shows "map\<^sub>0 a = G.map\<^sub>0 (F.map\<^sub>0 a)"
	using assms map\<^sub>0_def by auto

	lemma unit_char':
	assumes "B.obj a"
	shows "unit a = G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a)"
	proof -
	have "G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a) = unit a"
	proof (intro unit_eqI [of a "G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a)"])
	show "B.obj a" by fact
	show "\<guillemotleft>G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a) : map\<^sub>0 a \<Rightarrow>\<^sub>D map a\<guillemotright>"
	using assms by auto
	show "D.iso (G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a))"
	by (simp add: D.isos_compose F.unit_char(2) G.unit_char(2) assms)
	show "(G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a)) \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] =
	(map \<i>\<^sub>B[a] \<cdot>\<^sub>D cmp (a, a)) \<cdot>\<^sub>D
	(G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a) \<star>\<^sub>D G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a))"
	proof -
	have 1: "cmp (a, a) = G (\<Phi>\<^sub>F (a, a)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F a, F a)"
	proof -
	have "cmp (a, a) = (G (F (a \<star>\<^sub>B a)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (a, a))) \<cdot>\<^sub>D \<Phi>\<^sub>G (F a, F a)"
	using assms cmp_def B.VV.ide_char B.VV.arr_char B.VV.dom_char B.VV.cod_char
	B.VxV.dom_char B.objE D.comp_assoc B.obj_simps
	by simp
	also have "... = G (\<Phi>\<^sub>F (a, a)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F a, F a)"
	using assms D.comp_cod_arr B.VV.arr_char B.VV.ide_char by auto
	finally show ?thesis by blast
	qed
	have "(map \<i>\<^sub>B[a] \<cdot>\<^sub>D cmp (a, a)) \<cdot>\<^sub>D
	(G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a) \<star>\<^sub>D G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a)) =
	map \<i>\<^sub>B[a] \<cdot>\<^sub>D G (\<Phi>\<^sub>F (a, a)) \<cdot>\<^sub>D
	(\<Phi>\<^sub>G (F a, F a) \<cdot>\<^sub>D (G (F.unit a) \<star>\<^sub>D G (F.unit a))) \<cdot>\<^sub>D
	(G.unit (F.map\<^sub>0 a) \<star>\<^sub>D G.unit (F.map\<^sub>0 a))"
	using assms 1 D.comp_assoc D.interchange by simp
	also have "... = (map \<i>\<^sub>B[a] \<cdot>\<^sub>D G (\<Phi>\<^sub>F (a, a)) \<cdot>\<^sub>D G (F.unit a \<star>\<^sub>C F.unit a)) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F.map\<^sub>0 a, F.map\<^sub>0 a) \<cdot>\<^sub>D
	(G.unit (F.map\<^sub>0 a) \<star>\<^sub>D G.unit (F.map\<^sub>0 a))"
	using assms G.\<Phi>.naturality [of "(F.unit a, F.unit a)"]
	C.VV.arr_char C.VV.dom_char C.VV.cod_char G.FF_def D.comp_assoc
	by simp
	also have "... = (G (F \<i>\<^sub>B[a] \<cdot>\<^sub>C \<Phi>\<^sub>F (a, a) \<cdot>\<^sub>C (F.unit a \<star>\<^sub>C F.unit a))) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F.map\<^sub>0 a, F.map\<^sub>0 a) \<cdot>\<^sub>D
	(G.unit (F.map\<^sub>0 a) \<star>\<^sub>D G.unit (F.map\<^sub>0 a))"
	proof -
	have "C.arr (F \<i>\<^sub>B[a] \<cdot>\<^sub>C \<Phi>\<^sub>F (a, a) \<cdot>\<^sub>C (F.unit a \<star>\<^sub>C F.unit a))"
	using assms B.VV.ide_char B.VV.arr_char F.cmp_in_hom(2)
	by (intro C.seqI' C.comp_in_homI) auto
	hence "map \<i>\<^sub>B[a] \<cdot>\<^sub>D G (\<Phi>\<^sub>F (a, a)) \<cdot>\<^sub>D G (F.unit a \<star>\<^sub>C F.unit a) =
	G (F \<i>\<^sub>B[a] \<cdot>\<^sub>C \<Phi>\<^sub>F (a, a) \<cdot>\<^sub>C (F.unit a \<star>\<^sub>C F.unit a))"
	by auto
	thus ?thesis by argo
	qed
	also have "... = G (F.unit a \<cdot>\<^sub>C \<i>\<^sub>C[F.map\<^sub>0 a]) \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F.map\<^sub>0 a, F.map\<^sub>0 a) \<cdot>\<^sub>D
	(G.unit (F.map\<^sub>0 a) \<star>\<^sub>D G.unit (F.map\<^sub>0 a))"
	using assms F.unit_char C.comp_assoc by simp
	also have "... = G (F.unit a) \<cdot>\<^sub>D (G \<i>\<^sub>C[F.map\<^sub>0 a] \<cdot>\<^sub>D
	\<Phi>\<^sub>G (F.map\<^sub>0 a, F.map\<^sub>0 a)) \<cdot>\<^sub>D
	(G.unit (F.map\<^sub>0 a) \<star>\<^sub>D G.unit (F.map\<^sub>0 a))"
	using assms D.comp_assoc by simp
	also have "... = (G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a)) \<cdot>\<^sub>D \<i>\<^sub>D[G.map\<^sub>0 (F.map\<^sub>0 a)]"
	using assms G.unit_char D.comp_assoc by simp
	also have "... = (G (F.unit a) \<cdot>\<^sub>D G.unit (F.map\<^sub>0 a)) \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a]"
	using assms map\<^sub>0_def by auto
	finally show ?thesis by simp
	qed
	qed
	thus ?thesis by simp
	qed

	end

	subsection "Restriction of Pseudofunctors"

	text \<open>
	In this section, we construct the restriction and corestriction of a pseudofunctor to
	a subbicategory of its domain and codomain, respectively.
	\<close>

	locale restricted_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: 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> +
	C': subbicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C Arr
	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> :: "'c * 'c \<Rightarrow> 'd"
	and Arr :: "'c \<Rightarrow> bool"
	begin

	abbreviation map
	where "map \<equiv> \<lambda>\<mu>. if C'.arr \<mu> then F \<mu> else D.null"

	abbreviation cmp
	where "cmp \<equiv> \<lambda>\<mu>\<nu>. if C'.VV.arr \<mu>\<nu> then \<Phi> \<mu>\<nu> else D.null"

	interpretation "functor" C'.comp V\<^sub>D map
	using C'.inclusion C'.arr_char C'.dom_char C'.cod_char C'.seq_char C'.comp_char
	C'.arr_dom C'.arr_cod
	apply (unfold_locales)
	apply auto
	by presburger

	interpretation weak_arrow_of_homs C'.comp C'.src C'.trg V\<^sub>D src\<^sub>D trg\<^sub>D map
	using C'.arrE C'.ide_src C'.ide_trg C'.inclusion C'.src_def C'.trg_def
	F.weakly_preserves_src F.weakly_preserves_trg
	by unfold_locales auto

	interpretation H\<^sub>D\<^sub>'oFF: composite_functor C'.VV.comp D.VV.comp V\<^sub>D FF
	\<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>D snd \<mu>\<nu>\<close>
	..
	interpretation FoH\<^sub>C\<^sub>': composite_functor C'.VV.comp C'.comp V\<^sub>D
	\<open>\<lambda>\<mu>\<nu>. C'.hcomp (fst \<mu>\<nu>) (snd \<mu>\<nu>)\<close> map
	..

	interpretation \<Phi>: natural_transformation C'.VV.comp V\<^sub>D H\<^sub>D\<^sub>'oFF.map FoH\<^sub>C\<^sub>'.map cmp
	using C'.arr_char C'.dom_char C'.cod_char C'.VV.arr_char C'.VV.dom_char C'.VV.cod_char
	FF_def C'.inclusion C'.dom_closed C'.cod_closed C'.src_def C'.trg_def
	C'.hcomp_def C'.hcomp_closed F.\<Phi>.is_natural_1 F.\<Phi>.is_natural_2
	C.VV.arr_char C.VV.dom_char C.VV.cod_char F.FF_def
	by unfold_locales auto

	interpretation \<Phi>: natural_isomorphism C'.VV.comp V\<^sub>D H\<^sub>D\<^sub>'oFF.map FoH\<^sub>C\<^sub>'.map cmp
	using C.VV.ide_char C.VV.arr_char C'.VV.ide_char C'.VV.arr_char C'.arr_char
	C'.src_def C'.trg_def C'.ide_char F.\<Phi>.components_are_iso
	by unfold_locales auto

	sublocale pseudofunctor C'.comp C'.hcomp C'.\<a> \<i>\<^sub>C C'.src C'.trg V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D
	map cmp
	using F.assoc_coherence C'.VVV.arr_char C'.VV.arr_char C'.arr_char C'.hcomp_def
	C'.src_def C'.trg_def C'.assoc_closed C'.hcomp_closed C'.ide_char
	by unfold_locales (simp add: C'.ide_char C'.src_def C'.trg_def)

	lemma is_pseudofunctor:
	shows "pseudofunctor C'.comp C'.hcomp C'.\<a> \<i>\<^sub>C C'.src C'.trg V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D map cmp"
	..

	lemma map\<^sub>0_simp [simp]:
	assumes "C'.obj a"
	shows "map\<^sub>0 a = F.map\<^sub>0 a"
	using assms map\<^sub>0_def C'.obj_char by auto

	lemma unit_char':
	assumes "C'.obj a"
	shows "F.unit a = unit a"
	using assms map\<^sub>0_simp C'.obj_char F.unit_in_hom(2) [of a] F.unit_char(2-3) \<i>_simps(1)
	apply (intro unit_eqI)
	apply auto
	by blast

	end

	text \<open>
	We define the corestriction construction only for the case of sub-bicategories
	determined by a set of objects of the ambient bicategory.
	There are undoubtedly more general constructions, but this one is adequate for our
	present needs.
	\<close>

	locale corestricted_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: 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> +
	D': subbicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<lambda>\<mu>. D.arr \<mu> \<and> Obj (src\<^sub>D \<mu>) \<and> Obj (trg\<^sub>D \<mu>)\<close>
	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> :: "'c * 'c \<Rightarrow> 'd"
	and Obj :: "'d \<Rightarrow> bool" +
	assumes preserves_arr: "\<And>\<mu>. C.arr \<mu> \<Longrightarrow> D'.arr (F \<mu>)"
	begin

	abbreviation map
	where "map \<equiv> F"

	abbreviation cmp
	where "cmp \<equiv> \<Phi>"

	interpretation "functor" V\<^sub>C D'.comp F
	using preserves_arr F.is_extensional D'.arr_char D'.dom_char D'.cod_char D'.comp_char
	by (unfold_locales) auto

	interpretation weak_arrow_of_homs V\<^sub>C src\<^sub>C trg\<^sub>C D'.comp D'.src D'.trg F
	proof
	fix \<mu>
	assume \<mu>: "C.arr \<mu>"
	obtain \<phi> where \<phi>: "\<guillemotleft>\<phi> : F (src\<^sub>C \<mu>) \<Rightarrow>\<^sub>D src\<^sub>D (F \<mu>)\<guillemotright> \<and> D.iso \<phi>"
	using \<mu> F.weakly_preserves_src by auto
	have 2: "D'.in_hom \<phi> (F (src\<^sub>C \<mu>)) (D'.src (F \<mu>))"
	using \<mu> \<phi> D'.arr_char D'.dom_char D'.cod_char D'.src_def D.vconn_implies_hpar(1-2)
	preserves_arr
	by (metis (no_types, lifting) C.src.preserves_arr D'.in_hom_char D'.src.preserves_arr
	D.arrI)
	moreover have "D'.iso \<phi>"
	using 2 \<phi> D'.iso_char D'.arr_char by fastforce
	ultimately show "D'.isomorphic (F (src\<^sub>C \<mu>)) (D'.src (F \<mu>))"
	using D'.isomorphic_def by auto
	obtain \<psi> where \<psi>: "\<guillemotleft>\<psi> : F (trg\<^sub>C \<mu>) \<Rightarrow>\<^sub>D trg\<^sub>D (F \<mu>)\<guillemotright> \<and> D.iso \<psi>"
	using \<mu> F.weakly_preserves_trg by auto
	have 2: "D'.in_hom \<psi> (F (trg\<^sub>C \<mu>)) (D'.trg (F \<mu>))"
	using \<mu> \<psi> D'.arr_char D'.dom_char D'.cod_char D'.trg_def D.vconn_implies_hpar(1-2)
	preserves_arr
	by (metis (no_types, lifting) C.trg.preserves_arr D'.in_hom_char D'.trg.preserves_arr
	D.arrI)
	moreover have "D'.iso \<psi>"
	using 2 \<psi> D'.iso_char D'.arr_char by fastforce
	ultimately show "D'.isomorphic (F (trg\<^sub>C \<mu>)) (D'.trg (F \<mu>))"
	using D'.isomorphic_def by auto
	qed

	interpretation H\<^sub>D\<^sub>'oFF: composite_functor C.VV.comp D'.VV.comp D'.comp FF
	\<open>\<lambda>\<mu>\<nu>. D'.hcomp (fst \<mu>\<nu>) (snd \<mu>\<nu>)\<close>
	..
	interpretation FoH\<^sub>C: composite_functor C.VV.comp V\<^sub>C D'.comp \<open>\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>C snd \<mu>\<nu>\<close>
	F
	..

	interpretation natural_transformation C.VV.comp D'.comp H\<^sub>D\<^sub>'oFF.map FoH\<^sub>C.map \<Phi>
	proof
	show "\<And>\<mu>\<nu>. \<not> C.VV.arr \<mu>\<nu> \<Longrightarrow> \<Phi> \<mu>\<nu> = D'.null"
	by (simp add: F.\<Phi>.is_extensional)
	fix \<mu>\<nu>
	assume \<mu>\<nu>: "C.VV.arr \<mu>\<nu>"
	have 1: "D'.arr (\<Phi> \<mu>\<nu>)"
	using \<mu>\<nu> D'.arr_char F.\<Phi>.is_natural_1 F.\<Phi>.components_are_iso
	by (metis (no_types, lifting) D.src_vcomp D.trg_vcomp FoH\<^sub>C.preserves_arr
	F.\<Phi>.preserves_reflects_arr)
	show "D'.dom (\<Phi> \<mu>\<nu>) = H\<^sub>D\<^sub>'oFF.map (C.VV.dom \<mu>\<nu>)"
	using 1 \<mu>\<nu> D'.dom_char C.VV.arr_char C.VV.dom_char F.FF_def FF_def D'.hcomp_def
	by simp
	show "D'.cod (\<Phi> \<mu>\<nu>) = FoH\<^sub>C.map (C.VV.cod \<mu>\<nu>)"
	using 1 \<mu>\<nu> D'.cod_char C.VV.arr_char F.FF_def FF_def D'.hcomp_def by simp
	show "D'.comp (FoH\<^sub>C.map \<mu>\<nu>) (\<Phi> (C.VV.dom \<mu>\<nu>)) = \<Phi> \<mu>\<nu>"
	using 1 \<mu>\<nu> D'.arr_char D'.comp_char C.VV.dom_char F.\<Phi>.is_natural_1
	C.VV.arr_dom D.src_vcomp D.trg_vcomp FoH\<^sub>C.preserves_arr F.\<Phi>.preserves_reflects_arr
	by (metis (mono_tags, lifting))
	show "D'.comp (\<Phi> (C.VV.cod \<mu>\<nu>)) (H\<^sub>D\<^sub>'oFF.map \<mu>\<nu>) = \<Phi> \<mu>\<nu>"
	proof -
	have "Obj (F.map\<^sub>0 (trg\<^sub>C (fst \<mu>\<nu>))) \<and> Obj (F.map\<^sub>0 (trg\<^sub>C (snd \<mu>\<nu>)))"
	using \<mu>\<nu> C.VV.arr_char
	by (metis (no_types, lifting) C.src_trg C.trg.preserves_reflects_arr D'.arr_char
	F.map\<^sub>0_def preserves_hseq)
	moreover have "Obj (F.map\<^sub>0 (src\<^sub>C (snd \<mu>\<nu>)))"
	using \<mu>\<nu> C.VV.arr_char
	by (metis (no_types, lifting) C.src.preserves_reflects_arr C.trg_src D'.arr_char
	F.map\<^sub>0_def preserves_hseq)
	ultimately show ?thesis
	using \<mu>\<nu> 1 D'.arr_char D'.comp_char D'.hseq_char C.VV.arr_char C.VV.cod_char
	C.VxV.cod_char FF_def F.FF_def D'.hcomp_char preserves_hseq
	apply simp
	using F.\<Phi>.is_natural_2 by force
	qed
	qed

	interpretation natural_isomorphism C.VV.comp D'.comp H\<^sub>D\<^sub>'oFF.map FoH\<^sub>C.map \<Phi>
	apply unfold_locales
	using D'.iso_char D'.arr_char by fastforce

	sublocale pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C D'.comp D'.hcomp D'.\<a> \<i>\<^sub>D D'.src D'.trg
	F \<Phi>
	proof
	fix f g h
	assume f: "C.ide f" and g: "C.ide g" and h: "C.ide h"
	and fg: "src\<^sub>C f = trg\<^sub>C g" and gh: "src\<^sub>C g = trg\<^sub>C h"
	have "D'.comp (F \<a>\<^sub>C[f, g, h]) (D'.comp (\<Phi> (f \<star>\<^sub>C g, h)) (D'.hcomp (\<Phi> (f, g)) (F h))) =
	F \<a>\<^sub>C[f, g, h] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h)"
	proof -
	have 1: "D'.arr (cmp (f, g) \<star>\<^sub>D map h)"
	by (metis (mono_tags, lifting) C.ideD(1) D'.arr_char D'.hcomp_closed
	F.\<Phi>.preserves_reflects_arr F.cmp_simps(1-2) F.preserves_hseq
	f fg g gh h preserves_reflects_arr)
	moreover have 2: "D.seq (cmp (f \<star>\<^sub>C g, h)) (cmp (f, g) \<star>\<^sub>D map h)"
	using 1 f g h fg gh D'.arr_char C.VV.arr_char C.VV.dom_char C.VV.cod_char F.FF_def
	by (intro D.seqI) auto
	moreover have "D'.arr (cmp (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (cmp (f, g) \<star>\<^sub>D map h))"
	using 1 2 D'.arr_char
	by (metis (no_types, lifting) D'.comp_char D'.seq_char D.seqE F.\<Phi>.preserves_reflects_arr
	preserves_reflects_arr)
	ultimately show ?thesis
	using f g h fg gh D'.dom_char D'.cod_char D'.comp_char D'.hcomp_def C.VV.arr_char
	apply simp
	by force
	qed
	also have "... = \<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h]"
	using f g h fg gh F.assoc_coherence [of f g h] by blast
	also have "... = D'.comp (\<Phi> (f, g \<star>\<^sub>C h))
	(D'.comp (D'.hcomp (F f) (\<Phi> (g, h))) (D'.\<a> (F f) (F g) (F h)))"
	proof -
	have "D.seq (map f \<star>\<^sub>D cmp (g, h)) \<a>\<^sub>D[map f, map g, map h]"
	using f g h fg gh C.VV.arr_char C.VV.dom_char C.VV.cod_char F.FF_def
	by (intro D.seqI) auto
	moreover have "D'.arr \<a>\<^sub>D[map f, map g, map h]"
	using f g h fg gh D'.arr_char preserves_arr by auto
	moreover have "D'.arr (map f \<star>\<^sub>D cmp (g, h))"
	using f g h fg gh
	by (metis (no_types, lifting) D'.arr_char D.seqE D.vseq_implies_hpar(1)
	D.vseq_implies_hpar(2) calculation(1-2))
	moreover have "D'.arr ((map f \<star>\<^sub>D cmp (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[map f, map g, map h])"
	using f g h fg gh
	by (metis (no_types, lifting) D'.arr_char D'.comp_closed D.seqE
	calculation(1-3))
	moreover have "D.seq (cmp (f, g \<star>\<^sub>C h))
	((map f \<star>\<^sub>D cmp (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[map f, map g, map h])"
	using f g h fg gh F.cmp_simps'(1) F.cmp_simps(4) F.cmp_simps(5) by auto
	ultimately show ?thesis
	using f g h fg gh C.VV.arr_char D'.VVV.arr_char D'.VV.arr_char D'.comp_char
	D'.hcomp_def
	by simp
	qed
	finally show "D'.comp (F \<a>\<^sub>C[f, g, h])
	(D'.comp (\<Phi> (f \<star>\<^sub>C g, h)) (D'.hcomp (\<Phi> (f, g)) (F h))) =
	D'.comp (\<Phi> (f, g \<star>\<^sub>C h))
	(D'.comp (D'.hcomp (F f) (\<Phi> (g, h))) (D'.\<a> (F f) (F g) (F h)))"
	by blast
	qed

	lemma is_pseudofunctor:
	shows "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C D'.comp D'.hcomp D'.\<a> \<i>\<^sub>D D'.src D'.trg F \<Phi>"
	..

	lemma map\<^sub>0_simp [simp]:
	assumes "C.obj a"
	shows "map\<^sub>0 a = F.map\<^sub>0 a"
	using assms map\<^sub>0_def D'.src_def by auto

	lemma unit_char':
	assumes "C.obj a"
	shows "F.unit a = unit a"
	proof (intro unit_eqI)
	show "C.obj a" by fact
	show 1: "D'.in_hom (F.unit a) (map\<^sub>0 a) (map a)"
	using D'.arr_char D'.in_hom_char assms unit_in_hom(2) by force
	show "D'.iso (F.unit a)"
	using assms D'.iso_char D'.arr_char F.unit_char(2)
	\<open>D'.in_hom (F.unit a) (map\<^sub>0 a) (map a)\<close>
	by auto
	show "D'.comp (F.unit a) \<i>\<^sub>D[map\<^sub>0 a] =
	D'.comp (D'.comp (map \<i>\<^sub>C[a]) (cmp (a, a)))
	(D'.hcomp (F.unit a) (F.unit a))"
	proof -
	have "D'.comp (F.unit a) \<i>\<^sub>D[map\<^sub>0 a] = F.unit a \<cdot>\<^sub>D \<i>\<^sub>D[src\<^sub>D (map a)]"
	using assms D'.comp_char D'.arr_char
	apply simp
	by (metis (no_types, lifting) C.obj_simps(1-2) F.preserves_src preserves_arr)
	also have "... = (map \<i>\<^sub>C[a] \<cdot>\<^sub>D cmp (a, a)) \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D F.unit a)"
	using assms F.unit_char(3) [of a] by auto
	also have "... = D'.comp (D'.comp (map \<i>\<^sub>C[a]) (cmp (a, a)))
	(D'.hcomp (F.unit a) (F.unit a))"
	proof -
	have "D'.arr (map \<i>\<^sub>C[a] \<cdot>\<^sub>D cmp (a, a))"
	using assms D'.comp_simp by auto
	moreover have "D.seq (map \<i>\<^sub>C[a] \<cdot>\<^sub>D cmp (a, a)) (F.unit a \<star>\<^sub>D F.unit a)"
	using assms C.VV.arr_char F.cmp_simps(4-5)
	by (intro D.seqI) auto
	ultimately show ?thesis
	by (metis (no_types, lifting) "1" D'.comp_eqI' D'.hcomp_eqI' D'.hseqI'
	D'.iso_is_arr D'.seq_char D'.vconn_implies_hpar(1-2)
	\<i>_simps(1) \<open>D'.iso (F.unit a)\<close> assms map\<^sub>0_simps(2-3))
	qed
	finally show ?thesis by blast
	qed
	qed

	end


	subsection "Equivalence Pseudofunctors"

	text \<open>
	In this section, we define ``equivalence pseudofunctors'', which are pseudofunctors
	that are locally fully faithful, locally essentially surjective, and biessentially
	surjective on objects. In a later section, we will show that a pseudofunctor is
	an equivalence pseudofunctor if and only if it can be extended to an equivalence
	of bicategories.

	The definition below requires that an equivalence pseudofunctor be (globally) faithful
	with respect to vertical composition. Traditional formulations do not consider a
	pseudofunctor as a single global functor, so we have to consider whether this condition
	is too strong. In fact, a pseudofunctor (as defined here) is locally faithful if and
	only if it is globally faithful.
	\<close>

	context pseudofunctor
	begin

	definition locally_faithful
	where "locally_faithful \<equiv>
	\<forall>f g \<mu> \<mu>'. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> \<guillemotleft>\<mu>' : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> F \<mu> = F \<mu>' \<longrightarrow> \<mu> = \<mu>'"

	lemma locally_faithful_iff_faithful:
	shows "locally_faithful \<longleftrightarrow> faithful_functor V\<^sub>C V\<^sub>D F"
	proof
	show "faithful_functor V\<^sub>C V\<^sub>D F \<Longrightarrow> locally_faithful"
	by (metis category.in_homE faithful_functor.is_faithful functor_axioms
	functor_def locally_faithful_def)
	show "locally_faithful \<Longrightarrow> faithful_functor V\<^sub>C V\<^sub>D F"
	proof -
	assume 1: "locally_faithful"
	show "faithful_functor V\<^sub>C V\<^sub>D F"
	proof
	fix \<mu> \<mu>'
	assume par: "C.par \<mu> \<mu>'" and eq: "F \<mu> = F \<mu>'"
	obtain f g where fg: "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> \<guillemotleft>\<mu>' : f \<Rightarrow>\<^sub>C g\<guillemotright>"
	using par by auto
	show "\<mu> = \<mu>'"
	using 1 fg locally_faithful_def eq by simp
	qed
	qed
	qed

	end

	text \<open>
	In contrast, it is not true that a pseudofunctor that is locally full is also
	globally full, because we can have \<open>\<guillemotleft>\<nu> : F h \<Rightarrow>\<^sub>D F k\<guillemotright>\<close> even if \<open>h\<close> and \<open>k\<close>
	are not in the same hom-category. So it would be a mistake to require that an
	equivalence functor be globally full.
	\<close>

	locale equivalence_pseudofunctor =
	pseudofunctor +
	faithful_functor V\<^sub>C V\<^sub>D F +
	assumes biessentially_surjective_on_objects:
	"D.obj a' \<Longrightarrow> \<exists>a. C.obj a \<and> D.equivalent_objects (map\<^sub>0 a) a'"
	and locally_essentially_surjective:
	"\<lbrakk> C.obj a; C.obj b; \<guillemotleft>g : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 b\<guillemotright>; D.ide g \<rbrakk> \<Longrightarrow>
	\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<and> C.ide f \<and> D.isomorphic (F f) g"
	and locally_full:
	"\<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> : F f \<Rightarrow>\<^sub>D F f'\<guillemotright> \<rbrakk> \<Longrightarrow>
	\<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C f'\<guillemotright> \<and> F \<mu> = \<nu>"
	begin

	lemma reflects_ide:
	assumes "C.endo \<mu>" and "D.ide (F \<mu>)"
	shows "C.ide \<mu>"
	using assms is_faithful [of "C.dom \<mu>" \<mu>] C.ide_char'
	by (metis C.arr_dom C.cod_dom C.dom_dom C.seqE D.ide_char preserves_dom)

	lemma reflects_iso:
	assumes "C.arr \<mu>" and "D.iso (F \<mu>)"
	shows "C.iso \<mu>"
	proof -
	obtain \<mu>' where \<mu>': "\<guillemotleft>\<mu>' : C.cod \<mu> \<Rightarrow>\<^sub>C C.dom \<mu>\<guillemotright> \<and> F \<mu>' = D.inv (F \<mu>)"
	using assms locally_full [of "C.cod \<mu>" "C.dom \<mu>" "D.inv (F \<mu>)"]
	D.inv_in_hom C.in_homE preserves_hom C.in_homI
	by auto
	have "C.inverse_arrows \<mu> \<mu>'"
	using assms \<mu>' reflects_ide
	apply (intro C.inverse_arrowsI)
	apply (metis C.cod_comp C.dom_comp C.ide_dom C.in_homE C.seqI D.comp_inv_arr'
	faithful_functor_axioms faithful_functor_def functor.preserves_ide
	as_nat_trans.preserves_comp_2 preserves_dom)
	by (metis C.cod_comp C.dom_comp C.ide_cod C.in_homE C.seqI D.comp_arr_inv'
	faithful_functor_axioms faithful_functor_def functor.preserves_ide
	preserves_cod as_nat_trans.preserves_comp_2)
	thus ?thesis by auto
	qed

	lemma reflects_isomorphic:
	assumes "C.ide f" and "C.ide f'" and "src\<^sub>C f = src\<^sub>C f'" and "trg\<^sub>C f = trg\<^sub>C f'"
	and "D.isomorphic (F f) (F f')"
	shows "C.isomorphic f f'"
	using assms C.isomorphic_def D.isomorphic_def locally_full reflects_iso C.arrI
	by metis

	lemma reflects_equivalence:
	assumes "C.ide f" and "C.ide g"
	and "\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright>" and "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright>"
	and "equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g)
	(D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))
	(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
	shows "equivalence_in_bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
	proof -
	interpret E': equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>F f\<close> \<open>F g\<close>
	\<open>D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)\<close>
	\<open>D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)\<close>
	using assms by auto
	show ?thesis
	proof
	show "C.ide f"
	using assms(1) by simp
	show "C.ide g"
	using assms(2) by simp
	show "\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright>"
	using assms(3) by simp
	show "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright>"
	using assms(4) by simp
	have 0: "src\<^sub>C f = trg\<^sub>C g \<and> src\<^sub>C g = trg\<^sub>C f"
	using \<open>\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright>\<close>
	by (metis C.hseqE C.ideD(1) C.ide_cod C.ide_dom C.in_homE assms(4))
	show "C.iso \<eta>"
	proof -
	have "D.iso (F \<eta>)"
	proof -
	have 1: "\<guillemotleft>D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) : map\<^sub>0 (src\<^sub>C f) \<Rightarrow>\<^sub>D F g \<star>\<^sub>D F f\<guillemotright>"
	using \<open>C.ide f\<close> \<open>C.ide g\<close> \<open>\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright>\<close>
	unit_char cmp_in_hom cmp_components_are_iso
	by (metis (mono_tags, lifting) C.ideD(1) E'.unit_in_vhom preserves_src)
	have 2: "D.iso (\<Phi> (g, f)) \<and> \<guillemotleft>\<Phi> (g, f) : F g \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F (g \<star>\<^sub>C f)\<guillemotright>"
	using 0 \<open>C.ide f\<close> \<open>C.ide g\<close> cmp_in_hom by simp
	have 3: "D.iso (D.inv (unit (src\<^sub>C f))) \<and>
	\<guillemotleft>D.inv (unit (src\<^sub>C f)) : F (src\<^sub>C f) \<Rightarrow>\<^sub>D map\<^sub>0 (src\<^sub>C f)\<guillemotright>"
	using \<open>C.ide f\<close> unit_char by simp
	have "D.iso (\<Phi> (g, f) \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) \<cdot>\<^sub>D
	D.inv (unit (src\<^sub>C f)))"
	using 1 2 3 E'.unit_is_iso D.isos_compose by blast
	moreover have "\<Phi> (g, f) \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) \<cdot>\<^sub>D
	D.inv (unit (src\<^sub>C f)) =
	F \<eta>"
	proof -
	have "\<Phi> (g, f) \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) \<cdot>\<^sub>D
	D.inv (unit (src\<^sub>C f))
	= (\<Phi> (g, f) \<cdot>\<^sub>D (D.inv (\<Phi> (g, f))) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D (unit (src\<^sub>C f)) \<cdot>\<^sub>D
	D.inv (unit (src\<^sub>C f)))"
	using D.comp_assoc by simp
	also have "... = F (g \<star>\<^sub>C f) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D F (src\<^sub>C f)"
	using 2 3 D.comp_arr_inv D.comp_inv_arr D.inv_is_inverse
	by (metis C.ideD(1) C.obj_src D.comp_assoc D.dom_inv D.in_homE unit_char(2)
	assms(1))
	also have "... = F \<eta>"
	using \<open>\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright>\<close> D.comp_arr_dom D.comp_cod_arr by auto
	finally show ?thesis by simp
	qed
	ultimately show ?thesis by simp
	qed
	thus ?thesis
	using assms reflects_iso by auto
	qed
	show "C.iso \<epsilon>"
	proof -
	have "D.iso (F \<epsilon>)"
	proof -
	have 1: "\<guillemotleft>D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) : F f \<star>\<^sub>D F g \<Rightarrow>\<^sub>D map\<^sub>0 (src\<^sub>C g)\<guillemotright>"
	using \<open>C.ide f\<close> \<open>C.ide g\<close> \<open>\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright>\<close>
	unit_char cmp_in_hom cmp_components_are_iso
	by (metis (mono_tags, lifting) C.ideD(1) E'.counit_in_vhom preserves_src)
	have 2: "D.iso (D.inv (\<Phi> (f, g))) \<and>
	\<guillemotleft>D.inv (\<Phi> (f, g)) : F (f \<star>\<^sub>C g) \<Rightarrow>\<^sub>D F f \<star>\<^sub>D F g\<guillemotright>"
	using 0 \<open>C.ide f\<close> \<open>C.ide g\<close> \<open>\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright>\<close> cmp_in_hom(2) D.inv_in_hom
	by simp
	have 3: "D.iso (unit (trg\<^sub>C f)) \<and> \<guillemotleft>unit (trg\<^sub>C f) : map\<^sub>0 (trg\<^sub>C f) \<Rightarrow>\<^sub>D F (trg\<^sub>C f)\<guillemotright>"
	using \<open>C.ide f\<close> unit_char by simp
	have "D.iso (unit (trg\<^sub>C f) \<cdot>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D
	D.inv (\<Phi> (f, g)))"
	using 0 1 2 3 E'.counit_is_iso D.isos_compose
	by (metis D.arrI D.cod_comp D.cod_inv D.seqI D.seqI')
	moreover have "unit (trg\<^sub>C f) \<cdot>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D
	D.inv (\<Phi> (f, g)) =
	F \<epsilon>"
	proof -
	have "unit (trg\<^sub>C f) \<cdot>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D
	D.inv (\<Phi> (f, g)) =
	(unit (trg\<^sub>C f) \<cdot>\<^sub>D D.inv (unit (trg\<^sub>C f))) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D (\<Phi> (f, g) \<cdot>\<^sub>D D.inv (\<Phi> (f, g)))"
	using D.comp_assoc by simp
	also have "... = F (trg\<^sub>C f) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D F (f \<star>\<^sub>C g)"
	using 0 3 D.comp_arr_inv' D.comp_inv_arr'
	by (simp add: C.VV.arr_char C.VV.ide_char assms(1-2))
	also have "... = F \<epsilon>"
	using 0 \<open>\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright>\<close> D.comp_arr_dom D.comp_cod_arr by auto
	finally show ?thesis by simp
	qed
	ultimately show ?thesis by simp
	qed
	thus ?thesis
	using assms reflects_iso by auto
	qed
	qed
	qed

	lemma reflects_equivalence_map:
	assumes "C.ide f" and "D.equivalence_map (F f)"
	shows "C.equivalence_map f"
	proof -
	obtain g' \<phi> \<psi> where E': "equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) g' \<phi> \<psi>"
	using assms D.equivalence_map_def by auto
	interpret E': equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>F f\<close> g' \<phi> \<psi>
	using E' by auto
	obtain g where g: "\<guillemotleft>g : trg\<^sub>C f \<rightarrow>\<^sub>C src\<^sub>C f\<guillemotright> \<and> C.ide g \<and> D.isomorphic (F g) g'"
	using assms E'.antipar locally_essentially_surjective [of "trg\<^sub>C f" "src\<^sub>C f" g']
	by auto
	obtain \<mu> where \<mu>: "\<guillemotleft>\<mu> : g' \<Rightarrow>\<^sub>D F g\<guillemotright> \<and> D.iso \<mu>"
	using g D.isomorphic_def D.isomorphic_symmetric by blast
	interpret E'': equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>F f\<close> \<open>F g\<close>
	\<open>(F g \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D F f) \<cdot>\<^sub>D \<phi>\<close>
	\<open>\<psi> \<cdot>\<^sub>D (D.inv (F f) \<star>\<^sub>D g') \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv \<mu>)\<close>
	using assms \<mu> E'.equivalence_in_bicategory_axioms D.ide_is_iso
	D.equivalence_respects_iso [of "F f" g' \<phi> \<psi> "F f" "F f" \<mu> "F g"]
	by auto
	let ?\<eta>' = "\<Phi> (g, f) \<cdot>\<^sub>D (F g \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D F f) \<cdot>\<^sub>D \<phi> \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f))"
	have \<eta>': "\<guillemotleft>?\<eta>' : F (src\<^sub>C f) \<Rightarrow>\<^sub>D F (g \<star>\<^sub>C f)\<guillemotright>"
	using assms \<mu> g unit_char E'.unit_in_hom(2) E'.antipar E''.antipar cmp_in_hom
	apply (intro D.comp_in_homI)
	apply auto
	using E'.antipar(2) by blast
	have iso_\<eta>': "D.iso ?\<eta>'"
	using assms g \<mu> \<eta>' E'.antipar unit_char
	by (metis C.in_hhomE D.arrI D.inv_comp_left(2) D.inv_comp_right(2) D.iso_hcomp
	D.iso_inv_iso D.isos_compose D.seqE E''.antipar(2) E''.unit_is_iso
	E'.unit_is_iso as_nat_iso.components_are_iso cmp_components_are_iso)
	let ?\<epsilon>' = "unit (src\<^sub>C g) \<cdot>\<^sub>D \<psi> \<cdot>\<^sub>D (D.inv (F f) \<star>\<^sub>D g') \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv \<mu>) \<cdot>\<^sub>D
	D.inv (\<Phi> (f, g))"
	have \<epsilon>': "\<guillemotleft>?\<epsilon>' : F (f \<star>\<^sub>C g) \<Rightarrow>\<^sub>D F (trg\<^sub>C f)\<guillemotright>"
	proof (intro D.comp_in_homI)
	show "\<guillemotleft>D.inv (\<Phi> (f, g)) : F (f \<star>\<^sub>C g) \<Rightarrow>\<^sub>D F f \<star>\<^sub>D F g\<guillemotright>"
	using assms g cmp_in_hom C.VV.ide_char C.VV.arr_char by auto
	show "\<guillemotleft>F f \<star>\<^sub>D D.inv \<mu> : F f \<star>\<^sub>D F g \<Rightarrow>\<^sub>D F f \<star>\<^sub>D g'\<guillemotright>"
	using assms g \<mu> E''.antipar D.ide_in_hom(2) by auto
	show "\<guillemotleft>D.inv (F f) \<star>\<^sub>D g' : F f \<star>\<^sub>D g' \<Rightarrow>\<^sub>D F f \<star>\<^sub>D g'\<guillemotright>"
	using assms E'.antipar D.ide_is_iso by auto
	show "\<guillemotleft>\<psi> : F f \<star>\<^sub>D g' \<Rightarrow>\<^sub>D trg\<^sub>D (F f)\<guillemotright>"
	using E'.counit_in_hom by simp
	show "\<guillemotleft>unit (src\<^sub>C g) : trg\<^sub>D (F f) \<Rightarrow>\<^sub>D F (trg\<^sub>C f)\<guillemotright>"
	using assms g unit_char by auto
	qed
	have iso_\<epsilon>': "D.iso ?\<epsilon>'"
	proof -
	have "D.iso (\<Phi> (f, g))"
	using assms g C.VV.ide_char C.VV.arr_char by auto
	thus ?thesis
	by (metis C.in_hhomE D.arrI D.hseq_char' D.ide_is_iso D.inv_comp_left(2)
	D.inv_comp_right(2) D.iso_hcomp D.iso_inv_iso D.isos_compose D.seqE
	D.seq_if_composable E''.counit_is_iso E'.counit_is_iso E'.ide_left
	\<epsilon>' \<mu> g unit_char(2))
	qed
	obtain \<eta> where \<eta>: "\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright> \<and> F \<eta> = ?\<eta>'"
	using assms g E'.antipar \<eta>' locally_full [of "src\<^sub>C f" "g \<star>\<^sub>C f" ?\<eta>']
	by (metis C.ide_hcomp C.ideD(1) C.in_hhomE C.src.preserves_ide C.hcomp_simps(1-2)
	C.src_trg C.trg_trg)
	have iso_\<eta>: "C.iso \<eta>"
	using \<eta> \<eta>' iso_\<eta>' reflects_iso by auto
	have 1: "\<exists>\<epsilon>. \<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright> \<and> F \<epsilon> = ?\<epsilon>'"
	using assms g \<epsilon>' locally_full [of "f \<star>\<^sub>C g" "src\<^sub>C g" ?\<epsilon>'] by force
	obtain \<epsilon> where \<epsilon>: "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright> \<and> F \<epsilon> = ?\<epsilon>'"
	using 1 by blast
	have iso_\<epsilon>: "C.iso \<epsilon>"
	using \<epsilon> \<epsilon>' iso_\<epsilon>' reflects_iso by auto
	have "equivalence_in_bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
	using assms g \<eta> \<epsilon> iso_\<eta> iso_\<epsilon> by (unfold_locales, auto)
	thus ?thesis
	using C.equivalence_map_def by auto
	qed

	lemma reflects_equivalent_objects:
	assumes "C.obj a" and "C.obj b" and "D.equivalent_objects (map\<^sub>0 a) (map\<^sub>0 b)"
	shows "C.equivalent_objects a b"
	proof -
	obtain f' where f': "\<guillemotleft>f' : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 b\<guillemotright> \<and> D.equivalence_map f'"
	using assms D.equivalent_objects_def D.equivalence_map_def by auto
	obtain f where f: "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<and> C.ide f \<and> D.isomorphic (F f) f'"
	using assms f' locally_essentially_surjective [of a b f'] D.equivalence_map_is_ide
	by auto
	have "D.equivalence_map (F f)"
	using f f' D.equivalence_map_preserved_by_iso [of f' "F f"] D.isomorphic_symmetric
	by simp
	hence "C.equivalence_map f"
	using f f' reflects_equivalence_map [of f] by simp
	thus ?thesis
	using f C.equivalent_objects_def by auto
	qed

	end

	text\<open>
	For each pair of objects \<open>a\<close>, \<open>b\<close> of \<open>C\<close>, an equivalence pseudofunctor restricts
	to an equivalence of categories between \<open>C.hhom a b\<close> and \<open>D.hhom (map\<^sub>0 a) (map\<^sub>0 b)\<close>.
	\<close>

	(* TODO: Change the "perspective" of this locale to be the defined functor. *)
	locale equivalence_pseudofunctor_at_hom =
	equivalence_pseudofunctor +
	fixes a :: 'a and a' :: 'a
	assumes obj_a: "C.obj a"
	and obj_a': "C.obj a'"
	begin

	sublocale hhom\<^sub>C: subcategory V\<^sub>C \<open>\<lambda>\<mu>. \<guillemotleft>\<mu> : a \<rightarrow>\<^sub>C a'\<guillemotright>\<close>
	using C.hhom_is_subcategory by simp
	sublocale hhom\<^sub>D: subcategory V\<^sub>D \<open>\<lambda>\<mu>. \<guillemotleft>\<mu> : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a'\<guillemotright>\<close>
	using D.hhom_is_subcategory by simp

	definition F\<^sub>1
	where "F\<^sub>1 = (\<lambda>\<mu>. if hhom\<^sub>C.arr \<mu> then F \<mu> else D.null)"

	interpretation F\<^sub>1: "functor" hhom\<^sub>C.comp hhom\<^sub>D.comp F\<^sub>1
	unfolding F\<^sub>1_def
	using hhom\<^sub>C.arr_char hhom\<^sub>D.arr_char hhom\<^sub>C.dom_char hhom\<^sub>D.dom_char
	hhom\<^sub>C.cod_char hhom\<^sub>D.cod_char hhom\<^sub>C.seq_char hhom\<^sub>C.comp_char hhom\<^sub>D.comp_char
	by unfold_locales auto

	interpretation F\<^sub>1: fully_faithful_and_essentially_surjective_functor
	hhom\<^sub>C.comp hhom\<^sub>D.comp F\<^sub>1
	proof
	show "\<And>\<mu> \<mu>'. \<lbrakk>hhom\<^sub>C.par \<mu> \<mu>'; F\<^sub>1 \<mu> = F\<^sub>1 \<mu>'\<rbrakk> \<Longrightarrow> \<mu> = \<mu>'"
	unfolding F\<^sub>1_def
	using is_faithful hhom\<^sub>C.dom_char hhom\<^sub>D.dom_char hhom\<^sub>C.cod_char hhom\<^sub>D.cod_char
	by (metis C.in_hhom_def hhom\<^sub>C.arrE)
	show "\<And>f f' \<nu>. \<lbrakk>hhom\<^sub>C.ide f; hhom\<^sub>C.ide f'; hhom\<^sub>D.in_hom \<nu> (F\<^sub>1 f') (F\<^sub>1 f)\<rbrakk>
	\<Longrightarrow> \<exists>\<mu>. hhom\<^sub>C.in_hom \<mu> f' f \<and> F\<^sub>1 \<mu> = \<nu>"
	proof (unfold F\<^sub>1_def)
	fix f f' \<nu>
	assume f: "hhom\<^sub>C.ide f" and f': "hhom\<^sub>C.ide f'"
	assume "hhom\<^sub>D.in_hom \<nu> (if hhom\<^sub>C.arr f' then F f' else D.null)
	(if hhom\<^sub>C.arr f then F f else D.null)"
	hence \<nu>: "hhom\<^sub>D.in_hom \<nu> (F f') (F f)"
	using f f' by simp
	have "\<exists>\<mu>. hhom\<^sub>C.in_hom \<mu> f' f \<and> F \<mu> = \<nu>"
	proof -
	have 1: "src\<^sub>C f' = src\<^sub>C f \<and> trg\<^sub>C f' = trg\<^sub>C f"
	using f f' hhom\<^sub>C.ide_char by (metis C.in_hhomE hhom\<^sub>C.arrE)
	hence ex: "\<exists>\<mu>. C.in_hom \<mu> f' f \<and> F \<mu> = \<nu>"
	by (meson \<nu> f f' hhom\<^sub>D.in_hom_char horizontal_homs.hhom_is_subcategory
	locally_full subcategory.ide_char weak_arrow_of_homs_axioms
	weak_arrow_of_homs_def)
	obtain \<mu> where \<mu>: "C.in_hom \<mu> f' f \<and> F \<mu> = \<nu>"
	using ex by blast
	have "hhom\<^sub>C.in_hom \<mu> f' f"
	by (metis C.arrI C.in_hhom_def C.vconn_implies_hpar(1-2) \<mu> f f'
	hhom\<^sub>C.arr_char hhom\<^sub>C.ide_char hhom\<^sub>C.in_hom_char)
	thus ?thesis
	using \<mu> by auto
	qed
	thus "\<exists>\<mu>. hhom\<^sub>C.in_hom \<mu> f' f \<and> (if hhom\<^sub>C.arr \<mu> then F \<mu> else D.null) = \<nu>"
	by auto
	qed
	show "\<And>g. hhom\<^sub>D.ide g \<Longrightarrow> \<exists>f. hhom\<^sub>C.ide f \<and> hhom\<^sub>D.isomorphic (F\<^sub>1 f) g"
	proof (unfold F\<^sub>1_def)
	fix g
	assume g: "hhom\<^sub>D.ide g"
	show "\<exists>f. hhom\<^sub>C.ide f \<and> hhom\<^sub>D.isomorphic (if hhom\<^sub>C.arr f then F f else D.null) g"
	proof -
	have "C.obj a \<and> C.obj a'"
	using obj_a obj_a' by simp
	moreover have 1: "D.ide g \<and> \<guillemotleft>g : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a'\<guillemotright>"
	using g obj_a obj_a' hhom\<^sub>D.ide_char by auto
	ultimately have 2: "\<exists>f. C.in_hhom f a a' \<and> C.ide f \<and> D.isomorphic (F f) g"
	using locally_essentially_surjective [of a a' g] by simp
	obtain f \<phi> where f: "C.in_hhom f a a' \<and> C.ide f \<and> D.in_hom \<phi> (F f) g \<and> D.iso \<phi>"
	using 2 by auto
	have "hhom\<^sub>C.ide f"
	using f hhom\<^sub>C.ide_char hhom\<^sub>C.arr_char by simp
	moreover have "hhom\<^sub>D.isomorphic (F f) g"
	proof -
	have "hhom\<^sub>D.arr \<phi> \<and> hhom\<^sub>D.arr (D.inv \<phi>)"
	by (metis 1 D.arrI D.in_hhom_def D.vconn_implies_hpar(1-4) D.inv_in_homI
	f hhom\<^sub>D.arrI)
	hence "hhom\<^sub>D.in_hom \<phi> (F f) g \<and> hhom\<^sub>D.iso \<phi>"
	by (metis D.in_homE f hhom\<^sub>D.cod_simp hhom\<^sub>D.dom_simp hhom\<^sub>D.in_homI hhom\<^sub>D.iso_char)
	thus ?thesis
	unfolding hhom\<^sub>D.isomorphic_def by blast
	qed
	ultimately show "\<exists>f. hhom\<^sub>C.ide f \<and>
	hhom\<^sub>D.isomorphic (if hhom\<^sub>C.arr f then F f else D.null) g"
	by force
	qed
	qed
	qed

	lemma equivalence_functor_F\<^sub>1:
	shows "fully_faithful_and_essentially_surjective_functor hhom\<^sub>C.comp hhom\<^sub>D.comp F\<^sub>1"
	and "equivalence_functor hhom\<^sub>C.comp hhom\<^sub>D.comp F\<^sub>1"
	..

	definition G\<^sub>1
	where "G\<^sub>1 = (SOME G. \<exists>\<eta>\<epsilon>.
	adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G F\<^sub>1 (fst \<eta>\<epsilon>) (snd \<eta>\<epsilon>))"

	lemma G\<^sub>1_props:
	assumes "C.obj a" and "C.obj a'"
	shows "\<exists>\<eta> \<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>"
	proof -
	have "\<exists>G. \<exists>\<eta>\<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G F\<^sub>1 (fst \<eta>\<epsilon>) (snd \<eta>\<epsilon>)"
	using F\<^sub>1.extends_to_adjoint_equivalence by simp
	hence "\<exists>\<eta>\<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 (fst \<eta>\<epsilon>) (snd \<eta>\<epsilon>)"
	unfolding G\<^sub>1_def
	using someI_ex
	[of "\<lambda>G. \<exists>\<eta>\<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G F\<^sub>1 (fst \<eta>\<epsilon>) (snd \<eta>\<epsilon>)"]
	by blast
	thus ?thesis by simp
	qed

	definition \<eta>
	where "\<eta> = (SOME \<eta>. \<exists>\<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>)"

	definition \<epsilon>
	where "\<epsilon> = (SOME \<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>)"

	lemma \<eta>\<epsilon>_props:
	shows "adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>"
	using obj_a obj_a' \<eta>_def \<epsilon>_def G\<^sub>1_props
	someI_ex [of "\<lambda>\<eta>. \<exists>\<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>"]
	someI_ex [of "\<lambda>\<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>"]
	by simp

	sublocale \<eta>\<epsilon>: adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>
	using \<eta>\<epsilon>_props by simp

	sublocale \<eta>\<epsilon>: meta_adjunction hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta>\<epsilon>.\<phi> \<eta>\<epsilon>.\<psi>
	using \<eta>\<epsilon>.induces_meta_adjunction by simp

	end

	context identity_pseudofunctor
	begin

	sublocale equivalence_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
	map cmp
	using B.isomorphic_reflexive B.arrI
	apply unfold_locales
	by (auto simp add: B.equivalent_objects_reflexive map\<^sub>0_def B.obj_simps)

	lemma is_equivalence_pseudofunctor:
	shows "equivalence_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B
	map cmp"
	..

	end

	locale composite_equivalence_pseudofunctor =
	composite_pseudofunctor +
	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 +
	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
	begin

	interpretation faithful_functor V\<^sub>B V\<^sub>D \<open>G o F\<close>
	using F.faithful_functor_axioms G.faithful_functor_axioms faithful_functors_compose
	by blast

	interpretation equivalence_pseudofunctor 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>G o F\<close> cmp
	proof
	show "\<And>c. D.obj c \<Longrightarrow> \<exists>a. B.obj a \<and> D.equivalent_objects (map\<^sub>0 a) c"
	proof -
	fix c
	assume c: "D.obj c"
	obtain b where b: "C.obj b \<and> D.equivalent_objects (G.map\<^sub>0 b) c"
	using c G.biessentially_surjective_on_objects by auto
	obtain a where a: "B.obj a \<and> C.equivalent_objects (F.map\<^sub>0 a) b"
	using b F.biessentially_surjective_on_objects by auto
	have "D.equivalent_objects (map\<^sub>0 a) c"
	using a b map\<^sub>0_def G.preserves_equivalent_objects D.equivalent_objects_transitive
	by fastforce
	thus "\<exists>a. B.obj a \<and> D.equivalent_objects (map\<^sub>0 a) c"
	using a by auto
	qed
	show "\<And>a a' h. \<lbrakk>B.obj a; B.obj a'; \<guillemotleft>h : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a'\<guillemotright>; D.ide h\<rbrakk>
	\<Longrightarrow> \<exists>f. B.in_hhom f a a' \<and> B.ide f \<and> D.isomorphic ((G \<circ> F) f) h"
	proof -
	fix a a' h
	assume a: "B.obj a" and a': "B.obj a'"
	and h_in_hom: "\<guillemotleft>h : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a'\<guillemotright>" and ide_h: "D.ide h"
	obtain g
	where g: "C.in_hhom g (F.map\<^sub>0 a) (F.map\<^sub>0 a') \<and> C.ide g \<and> D.isomorphic (G g) h"
	using a a' h_in_hom ide_h map\<^sub>0_def B.obj_simps
	G.locally_essentially_surjective [of "F.map\<^sub>0 a" "F.map\<^sub>0 a'" h]
	by auto
	obtain f where f: "B.in_hhom f a a' \<and> B.ide f \<and> C.isomorphic (F f) g"
	using a a' g F.locally_essentially_surjective by blast
	have "D.isomorphic ((G o F) f) h"
	by (metis D.isomorphic_transitive G.preserves_isomorphic comp_apply f g)
	thus "\<exists>f. B.in_hhom f a a' \<and> B.ide f \<and> D.isomorphic ((G \<circ> F) f) h"
	using f by auto
	qed
	show "\<And>f f' \<nu>. \<lbrakk>B.ide f; B.ide f'; src\<^sub>B f = src\<^sub>B f'; trg\<^sub>B f = trg\<^sub>B f';
	\<guillemotleft>\<nu> : (G \<circ> F) f \<Rightarrow>\<^sub>D (G \<circ> F) f'\<guillemotright>\<rbrakk>
	\<Longrightarrow> \<exists>\<tau>. \<guillemotleft>\<tau> : f \<rightarrow>\<^sub>B f'\<guillemotright> \<and> (G \<circ> F) \<tau> = \<nu>"
	proof -
	fix f f' \<nu>
	assume f: "B.ide f" and f': "B.ide f'"
	and src: "src\<^sub>B f = src\<^sub>B f'" and trg: "trg\<^sub>B f = trg\<^sub>B f'"
	and \<nu>: "\<guillemotleft>\<nu> : (G \<circ> F) f \<Rightarrow>\<^sub>D (G \<circ> F) f'\<guillemotright>"
	have \<nu>: "\<guillemotleft>\<nu> : G (F f) \<Rightarrow>\<^sub>D G (F f')\<guillemotright>"
	using \<nu> by simp
	have 1: "src\<^sub>C (F f) = src\<^sub>C (F f') \<and> trg\<^sub>C (F f) = trg\<^sub>C (F f')"
	using f f' src trg by simp
	have 2: "\<exists>\<mu>. \<guillemotleft>\<mu> : F f \<Rightarrow>\<^sub>C F f'\<guillemotright> \<and> G \<mu> = \<nu>"
	using f f' 1 \<nu> G.locally_full F.preserves_ide by simp
	obtain \<mu> where \<mu>: "\<guillemotleft>\<mu> : F f \<Rightarrow>\<^sub>C F f'\<guillemotright> \<and> G \<mu> = \<nu>"
	using 2 by auto
	obtain \<tau> where \<tau>: "\<guillemotleft>\<tau> : f \<rightarrow>\<^sub>B f'\<guillemotright> \<and> F \<tau> = \<mu>"
	using f f' src trg 2 \<mu> F.locally_full by blast
	show "\<exists>\<tau>. \<guillemotleft>\<tau> : f \<rightarrow>\<^sub>B f'\<guillemotright> \<and> (G \<circ> F) \<tau> = \<nu>"
	using \<mu> \<tau> by auto
	qed
	qed

	sublocale equivalence_pseudofunctor 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>G o F\<close> cmp ..

	lemma is_equivalence_pseudofunctor:
	shows "equivalence_pseudofunctor 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
	(G o F) cmp"
	..

	end

	end