Datasets:

hoskinson-center
/

proof-pile

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

	section "Adjunctions in a Bicategory"

	theory InternalAdjunction
	imports CanonicalIsos Strictness
	begin

	text \<open>
	An \emph{internal adjunction} in a bicategory is a four-tuple \<open>(f, g, \<eta>, \<epsilon>)\<close>,
	where \<open>f\<close> and \<open>g\<close> are antiparallel 1-cells and \<open>\<guillemotleft>\<eta> : src f \<Rightarrow> g \<star> f\<guillemotright>\<close> and
	\<open>\<guillemotleft>\<epsilon> : f \<star> g \<Rightarrow> src g\<guillemotright>\<close> are 2-cells, such that the familiar ``triangle''
	(or ``zig-zag'') identities are satisfied. We state the triangle identities
	in two equivalent forms, each of which is convenient in certain situations.
	\<close>

	locale adjunction_in_bicategory =
	adjunction_data_in_bicategory +
	assumes triangle_left: "(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
	and triangle_right: "(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
	begin

	lemma triangle_left':
	shows "\<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[f] = f"
	using triangle_left triangle_equiv_form by simp

	lemma triangle_right':
	shows "\<r>[g] \<cdot> (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<cdot> \<l>\<^sup>-\<^sup>1[g] = g"
	using triangle_right triangle_equiv_form by simp

	end

	text \<open>
	Internal adjunctions have a number of properties, which we now develop,
	that generalize those of ordinary adjunctions involving functors and
	natural transformations.
	\<close>

	context bicategory
	begin

	lemma adjunction_unit_determines_counit:
	assumes "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
	and "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>'"
	shows "\<epsilon> = \<epsilon>'"
	proof -
	interpret E: adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using assms(1) by auto
	interpret E': adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>'
	using assms(2) by auto
	text \<open>
	Note that since we want to prove the the result for an arbitrary bicategory,
	not just in for a strict bicategory, the calculation is a little more involved
	than one might expect from a treatment that suppresses canonical isomorphisms.
	\<close>
	have "\<epsilon> = \<epsilon> \<cdot> (f \<star> \<r>[g] \<cdot> (g \<star> \<epsilon>') \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<cdot> \<l>\<^sup>-\<^sup>1[g])"
	using E'.triangle_right' comp_arr_dom by simp
	also have "... = \<epsilon> \<cdot> (f \<star> \<r>[g]) \<cdot> (f \<star> g \<star> \<epsilon>') \<cdot> (f \<star> \<a>[g, f, g]) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	using E.antipar whisker_left by simp
	also have "... = \<epsilon> \<cdot> ((f \<star> \<r>[g]) \<cdot> (f \<star> g \<star> \<epsilon>')) \<cdot> (f \<star> \<a>[g, f, g]) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	using comp_assoc by simp
	also have "... = \<epsilon> \<cdot> \<r>[f \<star> g] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, src g] \<cdot> (f \<star> g \<star> \<epsilon>')) \<cdot>
	(f \<star> \<a>[g, f, g]) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	proof -
	have "f \<star> \<r>[g] = \<r>[f \<star> g] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, src g]"
	using E.antipar(1) runit_hcomp(3) by auto
	thus ?thesis
	using comp_assoc by simp
	qed
	also have "... = (\<epsilon> \<cdot> \<r>[f \<star> g]) \<cdot> ((f \<star> g) \<star> \<epsilon>') \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> g] \<cdot>
	(f \<star> \<a>[g, f, g]) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	using E.antipar E'.counit_in_hom assoc'_naturality [of f g \<epsilon>'] comp_assoc by simp
	also have "... = \<r>[trg f] \<cdot> ((\<epsilon> \<star> trg f) \<cdot> ((f \<star> g) \<star> \<epsilon>')) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> g] \<cdot>
	(f \<star> \<a>[g, f, g]) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	using E.antipar E.counit_in_hom runit_naturality [of \<epsilon>] comp_assoc by simp
	also have "... = (\<l>[src g] \<cdot> (src g \<star> \<epsilon>')) \<cdot> (\<epsilon> \<star> f \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> g] \<cdot>
	(f \<star> \<a>[g, f, g]) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	proof -
	have "(\<epsilon> \<star> trg f) \<cdot> ((f \<star> g) \<star> \<epsilon>') = (src g \<star> \<epsilon>') \<cdot> (\<epsilon> \<star> f \<star> g)"
	using E.antipar interchange E.counit_in_hom comp_arr_dom comp_cod_arr
	by (metis E'.counit_simps(1-3) E.counit_simps(1-3))
	thus ?thesis
	using E.antipar comp_assoc unitor_coincidence by simp
	qed
	also have "... = \<epsilon>' \<cdot> \<l>[f \<star> g] \<cdot> (\<epsilon> \<star> f \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> g] \<cdot>
	(f \<star> \<a>[g, f, g]) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	proof -
	have "\<l>[src g] \<cdot> (src g \<star> \<epsilon>') = \<epsilon>' \<cdot> \<l>[f \<star> g]"
	using E.antipar lunit_naturality [of \<epsilon>'] by simp
	thus ?thesis
	using comp_assoc by simp
	qed
	also have "... = \<epsilon>' \<cdot> (\<l>[f] \<star> g) \<cdot> (\<a>\<^sup>-\<^sup>1[trg f, f, g] \<cdot> (\<epsilon> \<star> f \<star> g)) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> g] \<cdot>
	(f \<star> \<a>[g, f, g]) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	using E.antipar lunit_hcomp comp_assoc by simp
	also have "... = \<epsilon>' \<cdot> (\<l>[f] \<star> g) \<cdot> ((\<epsilon> \<star> f) \<star> g) \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, f, g] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> g] \<cdot>
	(f \<star> \<a>[g, f, g])) \<cdot> (f \<star> \<eta> \<star> g) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	using E.antipar assoc'_naturality [of \<epsilon> f g] comp_assoc by simp
	also have "... = \<epsilon>' \<cdot> (\<l>[f] \<star> g) \<cdot> ((\<epsilon> \<star> f) \<star> g) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<star> g) \<cdot>
	(\<a>\<^sup>-\<^sup>1[f, g \<star> f, g] \<cdot> (f \<star> \<eta> \<star> g)) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	proof -
	have "\<a>\<^sup>-\<^sup>1[f \<star> g, f, g] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> g] \<cdot> (f \<star> \<a>[g, f, g]) =
	(\<a>\<^sup>-\<^sup>1[f, g, f] \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[f, g \<star> f, g]"
	using E.antipar iso_assoc' pentagon' comp_assoc
	invert_side_of_triangle(2)
	[of "\<a>\<^sup>-\<^sup>1[f \<star> g, f, g] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> g]"
	"(\<a>\<^sup>-\<^sup>1[f, g, f] \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[f, g \<star> f, g]" "f \<star> \<a>\<^sup>-\<^sup>1[g, f, g]"]
	by simp
	thus ?thesis
	using comp_assoc by simp
	qed
	also have "... = \<epsilon>' \<cdot> (\<l>[f] \<star> g) \<cdot> ((\<epsilon> \<star> f) \<star> g) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<star> g) \<cdot>
	((f \<star> \<eta>) \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[f, trg g, g] \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	using E.antipar assoc'_naturality [of f \<eta> g] comp_assoc by simp
	also have "... = \<epsilon>' \<cdot> (\<l>[f] \<star> g) \<cdot> ((\<epsilon> \<star> f) \<star> g) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<star> g) \<cdot>
	((f \<star> \<eta>) \<star> g) \<cdot> (\<r>\<^sup>-\<^sup>1[f] \<star> g)"
	proof -
	have "\<a>\<^sup>-\<^sup>1[f, trg g, g] \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g]) = \<r>\<^sup>-\<^sup>1[f] \<star> g"
	proof -
	have "\<r>\<^sup>-\<^sup>1[f] \<star> g = inv (\<r>[f] \<star> g)"
	using E.antipar by simp
	also have "... = inv ((f \<star> \<l>[g]) \<cdot> \<a>[f, trg g, g])"
	using E.antipar by (simp add: triangle)
	also have "... = \<a>\<^sup>-\<^sup>1[f, trg g, g] \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[g])"
	using E.antipar inv_comp by simp
	finally show ?thesis by simp
	qed
	thus ?thesis by simp
	qed
	also have "... = \<epsilon>' \<cdot> (\<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[f] \<star> g)"
	using E.antipar whisker_right by simp
	also have "... = \<epsilon>'"
	using E.triangle_left' comp_arr_dom by simp
	finally show ?thesis by simp
	qed

	end

	subsection "Adjoint Transpose"

	context adjunction_in_bicategory
	begin

	interpretation E: self_evaluation_map V H \<a> \<i> src trg ..
	notation E.eval ("\<lbrace>_\<rbrace>")

	text \<open>
	Just as for an ordinary adjunction between categories, an adjunction in a bicategory
	determines bijections between hom-sets. There are two versions of this relationship:
	depending on whether the transposition is occurring on the left (\emph{i.e.}~``output'')
	side or the right (\emph{i.e.}~``input'') side.
	\<close>

	definition trnl\<^sub>\<eta>
	where "trnl\<^sub>\<eta> v \<mu> \<equiv> (g \<star> \<mu>) \<cdot> \<a>[g, f, v] \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v]"

	definition trnl\<^sub>\<epsilon>
	where "trnl\<^sub>\<epsilon> u \<nu> \<equiv> \<l>[u] \<cdot> (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> \<nu>)"

	lemma adjoint_transpose_left:
	assumes "ide u" and "ide v" and "src f = trg v" and "src g = trg u"
	shows "trnl\<^sub>\<eta> v \<in> hom (f \<star> v) u \<rightarrow> hom v (g \<star> u)"
	and "trnl\<^sub>\<epsilon> u \<in> hom v (g \<star> u) \<rightarrow> hom (f \<star> v) u"
	and "\<guillemotleft>\<mu> : f \<star> v \<Rightarrow> u\<guillemotright> \<Longrightarrow> trnl\<^sub>\<epsilon> u (trnl\<^sub>\<eta> v \<mu>) = \<mu>"
	and "\<guillemotleft>\<nu> : v \<Rightarrow> g \<star> u\<guillemotright> \<Longrightarrow> trnl\<^sub>\<eta> v (trnl\<^sub>\<epsilon> u \<nu>) = \<nu>"
	and "bij_betw (trnl\<^sub>\<eta> v) (hom (f \<star> v) u) (hom v (g \<star> u))"
	and "bij_betw (trnl\<^sub>\<epsilon> u) (hom v (g \<star> u)) (hom (f \<star> v) u)"
	proof -
	show A: "trnl\<^sub>\<eta> v \<in> hom (f \<star> v) u \<rightarrow> hom v (g \<star> u)"
	using assms antipar trnl\<^sub>\<eta>_def by fastforce
	show B: "trnl\<^sub>\<epsilon> u \<in> hom v (g \<star> u) \<rightarrow> hom (f \<star> v) u"
	using assms antipar trnl\<^sub>\<epsilon>_def by fastforce
	show C: "\<And>\<mu>. \<guillemotleft>\<mu> : f \<star> v \<Rightarrow> u\<guillemotright> \<Longrightarrow> trnl\<^sub>\<epsilon> u (trnl\<^sub>\<eta> v \<mu>) = \<mu>"
	proof -
	fix \<mu>
	assume \<mu>: "\<guillemotleft>\<mu> : f \<star> v \<Rightarrow> u\<guillemotright>"
	have "trnl\<^sub>\<epsilon> u (trnl\<^sub>\<eta> v \<mu>) =
	\<l>[u] \<cdot> (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> (g \<star> \<mu>) \<cdot> \<a>[g, f, v] \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v])"
	using trnl\<^sub>\<eta>_def trnl\<^sub>\<epsilon>_def by simp
	also have "... = \<l>[u] \<cdot> (\<epsilon> \<star> u) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> g \<star> \<mu>)) \<cdot> (f \<star> \<a>[g, f, v]) \<cdot>
	(f \<star> \<eta> \<star> v) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[v])"
	using assms \<mu> antipar whisker_left comp_assoc by auto
	also have "... = \<l>[u] \<cdot> ((\<epsilon> \<star> u) \<cdot> ((f \<star> g) \<star> \<mu>)) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> v] \<cdot> (f \<star> \<a>[g, f, v]) \<cdot>
	(f \<star> \<eta> \<star> v) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[v])"
	using assms \<mu> antipar assoc'_naturality [of f g \<mu>] comp_assoc by fastforce
	also have "... = \<l>[u] \<cdot> (trg u \<star> \<mu>) \<cdot>
	(\<epsilon> \<star> f \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> v] \<cdot> (f \<star> \<a>[g, f, v]) \<cdot>
	(f \<star> \<eta> \<star> v) \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[v])"
	proof -
	have "(\<epsilon> \<star> u) \<cdot> ((f \<star> g) \<star> \<mu>) = (trg u \<star> \<mu>) \<cdot> (\<epsilon> \<star> f \<star> v)"
	using assms \<mu> antipar comp_cod_arr comp_arr_dom
	interchange [of "trg u" \<epsilon> \<mu> "f \<star> v"] interchange [of \<epsilon> "f \<star> g" u \<mu>]
	by auto
	thus ?thesis
	using comp_assoc by simp
	qed
	also have "... = \<l>[u] \<cdot> (trg u \<star> \<mu>) \<cdot> \<a>[trg f, f, v] \<cdot>
	((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<star> v) \<cdot>
	\<a>\<^sup>-\<^sup>1[f, trg v, v] \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[v])"
	proof -
	have 1: "(\<epsilon> \<star> f \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> v] \<cdot> (f \<star> \<a>[g, f, v]) \<cdot> (f \<star> \<eta> \<star> v) =
	\<a>[trg f, f, v] \<cdot> ((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, trg v, v]"
	proof -
	have "(\<epsilon> \<star> f \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> v] \<cdot> (f \<star> \<a>[g, f, v]) \<cdot> (f \<star> \<eta> \<star> v) =
	(\<epsilon> \<star> f \<star> v) \<cdot>
	\<a>[f \<star> g, f, v] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, g \<star> f, v] \<cdot>
	(f \<star> \<eta> \<star> v)"
	proof -
	have "(\<epsilon> \<star> f \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f \<star> v] \<cdot> (f \<star> \<a>[g, f, v]) \<cdot> (f \<star> \<eta> \<star> v) =
	(\<epsilon> \<star> f \<star> v) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f \<star> v] \<cdot> (f \<star> \<a>[g, f, v])) \<cdot> (f \<star> \<eta> \<star> v)"
	using comp_assoc by simp
	also have "... = (\<epsilon> \<star> f \<star> v) \<cdot>
	\<a>[f \<star> g, f, v] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, g \<star> f, v] \<cdot>
	(f \<star> \<eta> \<star> v)"
	proof -
	have "\<a>\<^sup>-\<^sup>1[f, g, f \<star> v] \<cdot> (f \<star> \<a>[g, f, v]) =
	\<a>[f \<star> g, f, v] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, g \<star> f, v]"
	using assms antipar canI_associator_0 whisker_can_left_0 whisker_can_right_0
	canI_associator_hcomp(1-3)
	by simp
	thus ?thesis
	using comp_assoc by simp
	qed
	finally show ?thesis by blast
	qed
	also have "... = ((\<epsilon> \<star> f \<star> v) \<cdot> \<a>[f \<star> g, f, v]) \<cdot>
	(\<a>\<^sup>-\<^sup>1[f, g, f] \<star> v) \<cdot> ((f \<star> \<eta>) \<star> v) \<cdot>
	\<a>\<^sup>-\<^sup>1[f, trg v, v]"
	using assms \<mu> antipar assoc'_naturality [of f \<eta> v] comp_assoc by simp
	also have "... = (\<a>[trg f, f, v] \<cdot> ((\<epsilon> \<star> f) \<star> v)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<star> v) \<cdot> ((f \<star> \<eta>) \<star> v) \<cdot>
	\<a>\<^sup>-\<^sup>1[f, trg v, v]"
	using assms \<mu> antipar assoc_naturality [of \<epsilon> f v] by simp
	also have "... = \<a>[trg f, f, v] \<cdot>
	(((\<epsilon> \<star> f) \<star> v) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<star> v) \<cdot> ((f \<star> \<eta>) \<star> v)) \<cdot>
	\<a>\<^sup>-\<^sup>1[f, trg v, v]"
	using comp_assoc by simp
	also have "... = \<a>[trg f, f, v] \<cdot> ((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, trg v, v]"
	using assms \<mu> antipar whisker_right by simp
	finally show ?thesis by simp
	qed
	show ?thesis
	using 1 comp_assoc by metis
	qed
	also have "... = \<l>[u] \<cdot> (trg u \<star> \<mu>) \<cdot>
	\<a>[trg f, f, v] \<cdot> (\<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f] \<star> v) \<cdot> \<a>\<^sup>-\<^sup>1[f, trg v, v] \<cdot> (f \<star> \<l>\<^sup>-\<^sup>1[v])"
	using assms \<mu> antipar triangle_left by simp
	also have "... = \<l>[u] \<cdot> (trg u \<star> \<mu>) \<cdot> can (\<^bold>\<langle>trg u\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>v\<^bold>\<rangle>) (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>v\<^bold>\<rangle>)"
	using assms \<mu> antipar canI_unitor_0 canI_associator_1
	canI_associator_1(1-2) [of f v] whisker_can_right_0 whisker_can_left_0
	by simp
	also have "... = \<l>[u] \<cdot> (trg u \<star> \<mu>) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> v]"
	unfolding can_def using assms antipar comp_arr_dom comp_cod_arr \<ll>_ide_simp
	by simp
	also have "... = (\<l>[u] \<cdot> \<l>\<^sup>-\<^sup>1[u]) \<cdot> \<mu>"
	using assms \<mu> lunit'_naturality [of \<mu>] comp_assoc by auto
	also have "... = \<mu>"
	using assms \<mu> comp_cod_arr iso_lunit comp_arr_inv inv_is_inverse by auto
	finally show "trnl\<^sub>\<epsilon> u (trnl\<^sub>\<eta> v \<mu>) = \<mu>" by simp
	qed
	show D: "\<And>\<nu>. \<guillemotleft>\<nu> : v \<Rightarrow> g \<star> u\<guillemotright> \<Longrightarrow> trnl\<^sub>\<eta> v (trnl\<^sub>\<epsilon> u \<nu>) = \<nu>"
	proof -
	fix \<nu>
	assume \<nu>: "\<guillemotleft>\<nu> : v \<Rightarrow> g \<star> u\<guillemotright>"
	have "trnl\<^sub>\<eta> v (trnl\<^sub>\<epsilon> u \<nu>) =
	(g \<star> \<l>[u] \<cdot> (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> \<nu>)) \<cdot> \<a>[g, f, v] \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
	using trnl\<^sub>\<eta>_def trnl\<^sub>\<epsilon>_def by simp
	also have "... = (g \<star> \<l>[u]) \<cdot> (g \<star> \<epsilon> \<star> u) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> (g \<star> f \<star> \<nu>) \<cdot>
	\<a>[g, f, v] \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
	using assms \<nu> antipar interchange [of g "g \<cdot> g \<cdot> g"] comp_assoc by auto
	also have "... = ((g \<star> \<l>[u]) \<cdot> (g \<star> \<epsilon> \<star> u) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot>
	\<a>[g, f, g \<star> u] \<cdot> (\<eta> \<star> g \<star> u)) \<cdot> (trg v \<star> \<nu>) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
	proof -
	have "(g \<star> f \<star> \<nu>) \<cdot> \<a>[g, f, v] \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v] =
	\<a>[g, f, g \<star> u] \<cdot> (\<eta> \<star> g \<star> u) \<cdot> (trg v \<star> \<nu>) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
	proof -
	have "(g \<star> f \<star> \<nu>) \<cdot> \<a>[g, f, v] \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v] =
	\<a>[g, f, g \<star> u] \<cdot> ((g \<star> f) \<star> \<nu>) \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
	proof -
	have "(g \<star> f \<star> \<nu>) \<cdot> \<a>[g, f, v] = \<a>[g, f, g \<star> u] \<cdot> ((g \<star> f) \<star> \<nu>)"
	using assms \<nu> antipar assoc_naturality [of g f \<nu>] by auto
	thus ?thesis
	using assms comp_assoc by metis
	qed
	also have "... = \<a>[g, f, g \<star> u] \<cdot> (\<eta> \<star> g \<star> u) \<cdot> (trg v \<star> \<nu>) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
	proof -
	have "((g \<star> f) \<star> \<nu>) \<cdot> (\<eta> \<star> v) = (\<eta> \<star> g \<star> u) \<cdot> (trg v \<star> \<nu>)"
	using assms \<nu> antipar comp_arr_dom comp_cod_arr
	interchange [of "g \<star> f" \<eta> \<nu> v] interchange [of \<eta> "trg v" "g \<star> u" \<nu>]
	by auto
	thus ?thesis
	using comp_assoc by metis
	qed
	finally show ?thesis by blast
	qed
	thus ?thesis using comp_assoc by simp
	qed
	also have "... = \<l>[g \<star> u] \<cdot> (trg v \<star> \<nu>) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
	proof -
	have "(g \<star> \<l>[u]) \<cdot> (g \<star> \<epsilon> \<star> u) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> (\<eta> \<star> g \<star> u) =
	\<l>[g \<star> u]"
	proof -
	have "(g \<star> \<l>[u]) \<cdot> (g \<star> \<epsilon> \<star> u) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> (\<eta> \<star> g \<star> u) =
	(g \<star> \<l>[u]) \<cdot> \<a>[g, trg u, u] \<cdot>
	((g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<star> u) \<cdot>
	\<a>\<^sup>-\<^sup>1[trg v, g, u]"
	proof -
	have "(g \<star> \<l>[u]) \<cdot> (g \<star> \<epsilon> \<star> u) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> (\<eta> \<star> g \<star> u) =
	(g \<star> \<l>[u]) \<cdot> (g \<star> \<epsilon> \<star> u) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot>
	((\<eta> \<star> g \<star> u) \<cdot> \<a>[trg v, g, u]) \<cdot> \<a>\<^sup>-\<^sup>1[trg v, g, u]"
	using assms antipar comp_arr_dom comp_assoc comp_assoc_assoc'(1) by simp
	also have "... = (g \<star> \<l>[u]) \<cdot> (g \<star> \<epsilon> \<star> u) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot>
	(\<a>[g \<star> f, g, u] \<cdot> ((\<eta> \<star> g) \<star> u)) \<cdot> \<a>\<^sup>-\<^sup>1[trg v, g, u]"
	using assms antipar assoc_naturality [of \<eta> g u] by simp
	also have "... = (g \<star> \<l>[u]) \<cdot> (g \<star> \<epsilon> \<star> u) \<cdot>
	((g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> \<a>[g \<star> f, g, u]) \<cdot>
	((\<eta> \<star> g) \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[trg v, g, u]"
	using comp_assoc by simp
	also have "... = (g \<star> \<l>[u]) \<cdot> ((\<a>[g, trg u, u] \<cdot> \<a>\<^sup>-\<^sup>1[g, trg u, u]) \<cdot> (g \<star> \<epsilon> \<star> u)) \<cdot>
	((g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> \<a>[g \<star> f, g, u]) \<cdot>
	((\<eta> \<star> g) \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[trg v, g, u]"
	proof -
	have "(\<a>[g, trg u, u] \<cdot> \<a>\<^sup>-\<^sup>1[g, trg u, u]) \<cdot> (g \<star> \<epsilon> \<star> u) = g \<star> \<epsilon> \<star> u"
	using assms antipar comp_cod_arr comp_assoc_assoc'(1) by simp
	thus ?thesis
	using comp_assoc by simp
	qed
	also have "... = (g \<star> \<l>[u]) \<cdot> \<a>[g, trg u, u] \<cdot> (\<a>\<^sup>-\<^sup>1[g, trg u, u] \<cdot> (g \<star> \<epsilon> \<star> u)) \<cdot>
	(g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> \<a>[g \<star> f, g, u] \<cdot>
	((\<eta> \<star> g) \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[trg v, g, u]"
	using comp_assoc by simp
	also have "... = (g \<star> \<l>[u]) \<cdot> \<a>[g, trg u, u] \<cdot> (((g \<star> \<epsilon>) \<star> u) \<cdot> (\<a>\<^sup>-\<^sup>1[g, f \<star> g, u] \<cdot>
	(g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> \<a>[g \<star> f, g, u]) \<cdot>
	((\<eta> \<star> g) \<star> u)) \<cdot> \<a>\<^sup>-\<^sup>1[trg v, g, u]"
	using assms antipar assoc'_naturality [of g \<epsilon> u] comp_assoc by simp
	also have "... = (g \<star> \<l>[u]) \<cdot> \<a>[g, trg u, u] \<cdot>
	((g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<star> u) \<cdot>
	\<a>\<^sup>-\<^sup>1[trg v, g, u]"
	proof -
	have "\<a>\<^sup>-\<^sup>1[g, f \<star> g, u] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> \<a>[g \<star> f, g, u] =
	\<a>[g, f, g] \<star> u"
	using assms antipar canI_associator_0 whisker_can_left_0 whisker_can_right_0
	canI_associator_hcomp
	by simp
	hence "((g \<star> \<epsilon>) \<star> u) \<cdot>
	(\<a>\<^sup>-\<^sup>1[g, f \<star> g, u] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[f, g, u]) \<cdot> \<a>[g, f, g \<star> u] \<cdot> \<a>[g \<star> f, g, u]) \<cdot>
	((\<eta> \<star> g) \<star> u) =
	(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<star> u"
	using assms antipar whisker_right by simp
	thus ?thesis by simp
	qed
	finally show ?thesis by blast
	qed
	also have "... = (g \<star> \<l>[u]) \<cdot> \<a>[g, trg u, u] \<cdot> (\<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g] \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[trg g, g, u]"
	using assms antipar triangle_right by simp
	also have "... = can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>) (\<^bold>\<langle>trg g\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>)"
	proof -
	have "(g \<star> \<l>[u]) \<cdot> \<a>[g, trg u, u] \<cdot> (\<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g] \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[trg g, g, u] =
	((g \<star> \<l>[u]) \<cdot> \<a>[g, trg u, u] \<cdot> (\<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g] \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[trg g, g, u])"
	using comp_assoc by simp
	moreover have "... = can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>) (\<^bold>\<langle>trg g\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>)"
	using assms antipar canI_unitor_0 canI_associator_1 [of g u] inv_can
	whisker_can_left_0 whisker_can_right_0
	by simp
	ultimately show ?thesis by simp
	qed
	also have "... = \<l>[g \<star> u]"
	unfolding can_def using assms comp_arr_dom comp_cod_arr \<ll>_ide_simp by simp
	finally show ?thesis by simp
	qed
	thus ?thesis by simp
	qed
	also have "... = (\<l>[g \<star> u] \<cdot> \<l>\<^sup>-\<^sup>1[g \<star> u]) \<cdot> \<nu>"
	using assms \<nu> lunit'_naturality comp_assoc by auto
	also have "... = \<nu>"
	using assms \<nu> comp_cod_arr iso_lunit comp_arr_inv inv_is_inverse by auto
	finally show "trnl\<^sub>\<eta> v (trnl\<^sub>\<epsilon> u \<nu>) = \<nu>" by simp
	qed
	show "bij_betw (trnl\<^sub>\<eta> v) (hom (f \<star> v) u) (hom v (g \<star> u))"
	using A B C D by (intro bij_betwI) auto
	show "bij_betw (trnl\<^sub>\<epsilon> u) (hom v (g \<star> u)) (hom (f \<star> v) u)"
	using A B C D by (intro bij_betwI) auto
	qed

	lemma trnl\<^sub>\<epsilon>_comp:
	assumes "ide u" and "seq \<mu> \<nu>" and "src f = trg \<mu>"
	shows "trnl\<^sub>\<epsilon> u (\<mu> \<cdot> \<nu>) = trnl\<^sub>\<epsilon> u \<mu> \<cdot> (f \<star> \<nu>)"
	using assms trnl\<^sub>\<epsilon>_def whisker_left [of f \<mu> \<nu>] comp_assoc by auto

	definition trnr\<^sub>\<eta>
	where "trnr\<^sub>\<eta> v \<mu> \<equiv> (\<mu> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, g, f] \<cdot> (v \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[v]"

	definition trnr\<^sub>\<epsilon>
	where "trnr\<^sub>\<epsilon> u \<nu> \<equiv> \<r>[u] \<cdot> (u \<star> \<epsilon>) \<cdot> \<a>[u, f, g] \<cdot> (\<nu> \<star> g)"

	lemma adjoint_transpose_right:
	assumes "ide u" and "ide v" and "src v = trg g" and "src u = trg f"
	shows "trnr\<^sub>\<eta> v \<in> hom (v \<star> g) u \<rightarrow> hom v (u \<star> f)"
	and "trnr\<^sub>\<epsilon> u \<in> hom v (u \<star> f) \<rightarrow> hom (v \<star> g) u"
	and "\<guillemotleft>\<mu> : v \<star> g \<Rightarrow> u\<guillemotright> \<Longrightarrow> trnr\<^sub>\<epsilon> u (trnr\<^sub>\<eta> v \<mu>) = \<mu>"
	and "\<guillemotleft>\<nu> : v \<Rightarrow> u \<star> f\<guillemotright> \<Longrightarrow> trnr\<^sub>\<eta> v (trnr\<^sub>\<epsilon> u \<nu>) = \<nu>"
	and "bij_betw (trnr\<^sub>\<eta> v) (hom (v \<star> g) u) (hom v (u \<star> f))"
	and "bij_betw (trnr\<^sub>\<epsilon> u) (hom v (u \<star> f)) (hom (v \<star> g) u)"
	proof -
	show A: "trnr\<^sub>\<eta> v \<in> hom (v \<star> g) u \<rightarrow> hom v (u \<star> f)"
	using assms antipar trnr\<^sub>\<eta>_def by fastforce
	show B: "trnr\<^sub>\<epsilon> u \<in> hom v (u \<star> f) \<rightarrow> hom (v \<star> g) u"
	using assms antipar trnr\<^sub>\<epsilon>_def by fastforce
	show C: "\<And>\<mu>. \<guillemotleft>\<mu> : v \<star> g \<Rightarrow> u\<guillemotright> \<Longrightarrow> trnr\<^sub>\<epsilon> u (trnr\<^sub>\<eta> v \<mu>) = \<mu>"
	proof -
	fix \<mu>
	assume \<mu>: "\<guillemotleft>\<mu> : v \<star> g \<Rightarrow> u\<guillemotright>"
	have "trnr\<^sub>\<epsilon> u (trnr\<^sub>\<eta> v \<mu>) =
	\<r>[u] \<cdot> (u \<star> \<epsilon>) \<cdot> \<a>[u, f, g] \<cdot> ((\<mu> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, g, f] \<cdot> (v \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[v] \<star> g)"
	unfolding trnr\<^sub>\<epsilon>_def trnr\<^sub>\<eta>_def by simp
	also have "... = \<r>[u] \<cdot> (u \<star> \<epsilon>) \<cdot> (\<a>[u, f, g] \<cdot> ((\<mu> \<star> f) \<star> g)) \<cdot>
	(\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g) \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g)"
	using assms \<mu> antipar whisker_right comp_assoc by auto
	also have "... = \<r>[u] \<cdot> (u \<star> \<epsilon>) \<cdot> ((\<mu> \<star> f \<star> g) \<cdot> \<a>[v \<star> g, f, g]) \<cdot>
	(\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g) \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g)"
	using assms \<mu> antipar assoc_naturality [of \<mu> f g] by auto
	also have "... = \<r>[u] \<cdot> ((u \<star> \<epsilon>) \<cdot> (\<mu> \<star> f \<star> g)) \<cdot> \<a>[v \<star> g, f, g] \<cdot>
	(\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g) \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g)"
	using comp_assoc by auto
	also have "... = \<r>[u] \<cdot> (\<mu> \<star> src u) \<cdot> ((v \<star> g) \<star> \<epsilon>) \<cdot> \<a>[v \<star> g, f, g] \<cdot>
	(\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g) \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g)"
	proof -
	have "(u \<star> \<epsilon>) \<cdot> (\<mu> \<star> f \<star> g) = (\<mu> \<star> src u) \<cdot> ((v \<star> g) \<star> \<epsilon>)"
	using assms \<mu> antipar comp_arr_dom comp_cod_arr
	interchange [of \<mu> "v \<star> g" "src u" \<epsilon>] interchange [of u \<mu> \<epsilon> "f \<star> g"]
	by auto
	thus ?thesis
	using comp_assoc by simp
	qed
	also have "... = \<r>[u] \<cdot> (\<mu> \<star> src u) \<cdot>
	(((v \<star> g) \<star> \<epsilon>) \<cdot> \<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g)) \<cdot>
	(\<r>\<^sup>-\<^sup>1[v] \<star> g)"
	using comp_assoc by simp
	also have "... = \<r>[u] \<cdot> (\<mu> \<star> src u) \<cdot>
	(\<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> (v \<star> (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g)) \<cdot>
	\<a>[v, src v, g]) \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g)"
	proof -
	have "((v \<star> g) \<star> \<epsilon>) \<cdot> \<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g) =
	\<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> (v \<star> (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g)) \<cdot> \<a>[v, src v, g]"
	proof -
	have "((v \<star> g) \<star> \<epsilon>) \<cdot> \<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g) =
	((\<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> \<a>[v, g, src u]) \<cdot> ((v \<star> g) \<star> \<epsilon>)) \<cdot>
	\<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g)"
	proof -
	have "arr v \<and> dom v = v \<and> cod v = v"
	using assms(2) ide_char by blast
	moreover have "arr g \<and> dom g = g \<and> cod g = g"
	using ide_right ide_char by blast
	ultimately show ?thesis
	by (metis (no_types) antipar(2) assms(3-4) assoc_naturality
	counit_simps(1,3,5) hcomp_reassoc(1) comp_assoc)
	qed
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> (\<a>[v, g, src u] \<cdot> ((v \<star> g) \<star> \<epsilon>)) \<cdot>
	\<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> ((v \<star> \<eta>) \<star> g)"
	using comp_assoc by simp
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> ((v \<star> g \<star> \<epsilon>) \<cdot> \<a>[v, g, f \<star> g]) \<cdot>
	\<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot>
	(\<a>\<^sup>-\<^sup>1[v, g \<star> f, g] \<cdot> \<a>[v, g \<star> f, g]) \<cdot> ((v \<star> \<eta>) \<star> g)"
	proof -
	have "\<a>[v, g, src u] \<cdot> ((v \<star> g) \<star> \<epsilon>) = (v \<star> g \<star> \<epsilon>) \<cdot> \<a>[v, g, f \<star> g]"
	using assms antipar assoc_naturality [of v g \<epsilon>] by simp
	moreover have "(\<a>\<^sup>-\<^sup>1[v, g \<star> f, g] \<cdot> \<a>[v, g \<star> f, g]) \<cdot> ((v \<star> \<eta>) \<star> g) = (v \<star> \<eta>) \<star> g"
	using assms antipar comp_cod_arr comp_assoc_assoc'(2) by simp
	ultimately show ?thesis by simp
	qed
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> (v \<star> g \<star> \<epsilon>) \<cdot>
	\<a>[v, g, f \<star> g] \<cdot> \<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot>
	\<a>\<^sup>-\<^sup>1[v, g \<star> f, g] \<cdot> \<a>[v, g \<star> f, g] \<cdot> ((v \<star> \<eta>) \<star> g)"
	using comp_assoc by simp
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> ((v \<star> g \<star> \<epsilon>) \<cdot>
	(\<a>[v, g, f \<star> g] \<cdot> \<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot>
	\<a>\<^sup>-\<^sup>1[v, g \<star> f, g]) \<cdot> (v \<star> \<eta> \<star> g)) \<cdot> \<a>[v, src v, g]"
	using assms antipar assoc_naturality [of v \<eta> g] comp_assoc by simp
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot>
	((v \<star> g \<star> \<epsilon>) \<cdot> (v \<star> \<a>[g, f, g]) \<cdot> (v \<star> \<eta> \<star> g)) \<cdot>
	\<a>[v, src v, g]"
	proof -
	have "\<a>[v, g, f \<star> g] \<cdot> \<a>[v \<star> g, f, g] \<cdot> (\<a>\<^sup>-\<^sup>1[v, g, f] \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[v, g \<star> f, g] =
	v \<star> \<a>[g, f, g]"
	using assms antipar canI_associator_0 canI_associator_hcomp
	whisker_can_left_0 whisker_can_right_0
	by simp
	thus ?thesis
	using assms antipar whisker_left by simp
	qed
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot>
	(v \<star> (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g)) \<cdot>
	\<a>[v, src v, g]"
	using assms antipar whisker_left by simp
	finally show ?thesis by simp
	qed
	thus ?thesis by auto
	qed
	also have "... = \<r>[u] \<cdot> (\<mu> \<star> src u) \<cdot>
	\<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> (v \<star> \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]) \<cdot>
	\<a>[v, src v, g] \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g)"
	using triangle_right comp_assoc by simp
	also have "... = \<r>[u] \<cdot> (\<mu> \<star> src u) \<cdot> \<r>\<^sup>-\<^sup>1[v \<star> g]"
	proof -
	have "\<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> (v \<star> \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]) \<cdot> \<a>[v, src v, g] \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g) = \<r>\<^sup>-\<^sup>1[v \<star> g]"
	proof -
	have "\<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> (v \<star> \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]) \<cdot> \<a>[v, src v, g] \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g) =
	\<a>\<^sup>-\<^sup>1[v, g, trg f] \<cdot> can (\<^bold>\<langle>v\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>src g\<^bold>\<rangle>\<^sub>0) (\<^bold>\<langle>v\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>)"
	using assms canI_unitor_0 canI_associator_1(2-3) whisker_can_left_0(1)
	whisker_can_right_0
	by simp
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src g] \<cdot> can (\<^bold>\<langle>v\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>src g\<^bold>\<rangle>\<^sub>0) (\<^bold>\<langle>v\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>)"
	using antipar by simp
	(* TODO: There should be an alternate version of whisker_can_left for this. *)
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src g] \<cdot> (v \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>src g\<^bold>\<rangle>\<^sub>0) \<^bold>\<langle>g\<^bold>\<rangle>)"
	using assms canI_unitor_0(2) whisker_can_left_0 by simp
	also have "... = \<a>\<^sup>-\<^sup>1[v, g, src g] \<cdot> (v \<star> \<r>\<^sup>-\<^sup>1[g])"
	using assms canI_unitor_0(2) by simp
	also have "... = \<r>\<^sup>-\<^sup>1[v \<star> g]"
	using assms runit_hcomp(2) by simp
	finally have "\<a>\<^sup>-\<^sup>1[v, g, src u] \<cdot> (v \<star> \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]) \<cdot> \<a>[v, src v, g] \<cdot> (\<r>\<^sup>-\<^sup>1[v] \<star> g) =
	\<r>\<^sup>-\<^sup>1[v \<star> g]"
	by simp
	thus ?thesis by simp
	qed
	thus ?thesis by simp
	qed
	also have "... = (\<r>[u] \<cdot> \<r>\<^sup>-\<^sup>1[u]) \<cdot> \<mu>"
	using assms \<mu> runit'_naturality [of \<mu>] comp_assoc by auto
	also have "... = \<mu>"
	using \<mu> comp_cod_arr iso_runit inv_is_inverse comp_arr_inv by auto
	finally show "trnr\<^sub>\<epsilon> u (trnr\<^sub>\<eta> v \<mu>) = \<mu>" by simp
	qed
	show D: "\<And>\<nu>. \<guillemotleft>\<nu> : v \<Rightarrow> u \<star> f\<guillemotright> \<Longrightarrow> trnr\<^sub>\<eta> v (trnr\<^sub>\<epsilon> u \<nu>) = \<nu>"
	proof -
	fix \<nu>
	assume \<nu>: "\<guillemotleft>\<nu> : v \<Rightarrow> u \<star> f\<guillemotright>"
	have "trnr\<^sub>\<eta> v (trnr\<^sub>\<epsilon> u \<nu>) =
	(\<r>[u] \<cdot> (u \<star> \<epsilon>) \<cdot> \<a>[u, f, g] \<cdot> (\<nu> \<star> g) \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, g, f] \<cdot> (v \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	unfolding trnr\<^sub>\<eta>_def trnr\<^sub>\<epsilon>_def by simp
	also have "... = (\<r>[u] \<star> f) \<cdot> ((u \<star> \<epsilon>) \<star> f) \<cdot> (\<a>[u, f, g] \<star> f) \<cdot>
	(((\<nu> \<star> g) \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, g, f]) \<cdot> (v \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using assms \<nu> antipar whisker_right comp_assoc by auto
	also have "... = (\<r>[u] \<star> f) \<cdot> ((u \<star> \<epsilon>) \<star> f) \<cdot> (\<a>[u, f, g] \<star> f) \<cdot>
	(\<a>\<^sup>-\<^sup>1[u \<star> f, g, f] \<cdot> (\<nu> \<star> g \<star> f)) \<cdot> (v \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using assms \<nu> antipar assoc'_naturality [of \<nu> g f] by auto
	also have "... = (\<r>[u] \<star> f) \<cdot> ((u \<star> \<epsilon>) \<star> f) \<cdot> (\<a>[u, f, g] \<star> f) \<cdot>
	\<a>\<^sup>-\<^sup>1[u \<star> f, g, f] \<cdot> ((\<nu> \<star> g \<star> f) \<cdot> (v \<star> \<eta>)) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using comp_assoc by simp
	also have "... = (\<r>[u] \<star> f) \<cdot> ((u \<star> \<epsilon>) \<star> f) \<cdot> (\<a>[u, f, g] \<star> f) \<cdot>
	\<a>\<^sup>-\<^sup>1[u \<star> f, g, f] \<cdot> (((u \<star> f) \<star> \<eta>) \<cdot> (\<nu> \<star> src v)) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	proof -
	have "(\<nu> \<star> g \<star> f) \<cdot> (v \<star> \<eta>) = ((u \<star> f) \<star> \<eta>) \<cdot> (\<nu> \<star> src v)"
	using assms \<nu> antipar interchange [of "u \<star> f" \<nu> \<eta> "src v"]
	interchange [of \<nu> v "g \<star> f" \<eta>] comp_arr_dom comp_cod_arr
	by auto
	thus ?thesis by simp
	qed
	also have "... = ((\<r>[u] \<star> f) \<cdot> ((u \<star> \<epsilon>) \<star> f) \<cdot>
	((\<a>[u, f, g] \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> f, g, f]) \<cdot>
	((u \<star> f) \<star> \<eta>)) \<cdot> (\<nu> \<star> src v) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using comp_assoc by simp
	also have "... = ((\<r>[u] \<star> f) \<cdot> ((u \<star> \<epsilon>) \<star> f) \<cdot>
	(\<a>\<^sup>-\<^sup>1[u, f \<star> g, f] \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[f, g, f]) \<cdot> \<a>[u, f, g \<star> f]) \<cdot>
	((u \<star> f) \<star> \<eta>)) \<cdot> (\<nu> \<star> src v) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using assms antipar canI_associator_hcomp canI_associator_0 whisker_can_left_0
	whisker_can_right_0
	by simp
	also have "... = ((\<r>[u] \<star> f) \<cdot> (((u \<star> \<epsilon>) \<star> f) \<cdot>
	\<a>\<^sup>-\<^sup>1[u, f \<star> g, f]) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[f, g, f]) \<cdot> (\<a>[u, f, g \<star> f]) \<cdot>
	((u \<star> f) \<star> \<eta>)) \<cdot> (\<nu> \<star> src v) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using comp_assoc by simp
	also have "... = ((\<r>[u] \<star> f) \<cdot> (\<a>\<^sup>-\<^sup>1[u, src u, f] \<cdot> (u \<star> \<epsilon> \<star> f)) \<cdot>
	(u \<star> \<a>\<^sup>-\<^sup>1[f, g, f]) \<cdot> ((u \<star> f \<star> \<eta>) \<cdot> \<a>[u, f, src f])) \<cdot>
	(\<nu> \<star> src v) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using assms antipar assoc'_naturality [of u \<epsilon> f] assoc_naturality [of u f \<eta>]
	by auto
	also have "... = (\<r>[u] \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[u, src u, f] \<cdot>
	((u \<star> \<epsilon> \<star> f) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[f, g, f]) \<cdot> (u \<star> f \<star> \<eta>)) \<cdot> \<a>[u, f, src f] \<cdot>
	(\<nu> \<star> src v) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using comp_assoc by simp
	also have "... = (\<r>[u] \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[u, src u, f] \<cdot>
	(u \<star> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>)) \<cdot> \<a>[u, f, src f] \<cdot>
	(\<nu> \<star> src v) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using assms antipar whisker_left by auto
	also have "... = ((\<r>[u] \<star> f) \<cdot> (\<a>\<^sup>-\<^sup>1[u, src u, f] \<cdot> (u \<star> \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]) \<cdot> \<a>[u, f, src f])) \<cdot>
	(\<nu> \<star> src v) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	using assms antipar triangle_left comp_assoc by simp
	also have "... = \<r>[u \<star> f] \<cdot> (\<nu> \<star> src v) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
	proof -
	have "(\<r>[u] \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[u, src u, f] \<cdot> (u \<star> \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]) \<cdot> \<a>[u, f, src f] =
	((u \<star> \<l>[f]) \<cdot> (\<a>[u, src u, f] \<cdot> \<a>\<^sup>-\<^sup>1[u, src u, f])) \<cdot>
	(u \<star> \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]) \<cdot> \<a>[u, f, src f]"
	using assms ide_left ide_right antipar triangle comp_assoc by metis
	also have "... = (u \<star> \<r>[f]) \<cdot> \<a>[u, f, src f]"
	using assms antipar canI_associator_1 canI_unitor_0 whisker_can_left_0
	whisker_can_right_0 canI_associator_1
	by simp
	also have "... = \<r>[u \<star> f]"
	using assms antipar runit_hcomp by simp
	finally show ?thesis by simp
	qed
	also have "... = (\<r>[u \<star> f] \<cdot> \<r>\<^sup>-\<^sup>1[u \<star> f]) \<cdot> \<nu>"
	using assms \<nu> runit'_naturality [of \<nu>] comp_assoc by auto
	also have "... = \<nu>"
	using assms \<nu> comp_cod_arr comp_arr_inv inv_is_inverse iso_runit by auto
	finally show "trnr\<^sub>\<eta> v (trnr\<^sub>\<epsilon> u \<nu>) = \<nu>" by auto
	qed
	show "bij_betw (trnr\<^sub>\<eta> v) (hom (v \<star> g) u) (hom v (u \<star> f))"
	using A B C D by (intro bij_betwI, auto)
	show "bij_betw (trnr\<^sub>\<epsilon> u) (hom v (u \<star> f)) (hom (v \<star> g) u)"
	using A B C D by (intro bij_betwI, auto)
	qed

	lemma trnr\<^sub>\<eta>_comp:
	assumes "ide v" and "seq \<mu> \<nu>" and "src \<mu> = trg f"
	shows "trnr\<^sub>\<eta> v (\<mu> \<cdot> \<nu>) = (\<mu> \<star> f) \<cdot> trnr\<^sub>\<eta> v \<nu>"
	using assms trnr\<^sub>\<eta>_def whisker_right comp_assoc by auto

	end

	text \<open>
	It is useful to have at hand the simpler versions of the preceding results that
	hold in a normal bicategory and in a strict bicategory.
	\<close>

	locale adjunction_in_normal_bicategory =
	normal_bicategory +
	adjunction_in_bicategory
	begin

	lemma triangle_left:
	shows "(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = f"
	using triangle_left strict_lunit strict_runit by simp

	lemma triangle_right:
	shows "(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = g"
	using triangle_right strict_lunit strict_runit by simp

	lemma trnr\<^sub>\<eta>_eq:
	assumes "ide u" and "ide v"
	and "src v = trg g" and "src u = trg f"
	and "\<mu> \<in> hom (v \<star> g) u"
	shows "trnr\<^sub>\<eta> v \<mu> = (\<mu> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, g, f] \<cdot> (v \<star> \<eta>)"
	unfolding trnr\<^sub>\<eta>_def
	using assms antipar strict_runit' comp_arr_ide [of "\<r>\<^sup>-\<^sup>1[v]" "v \<star> \<eta>"] hcomp_arr_obj
	by auto

	lemma trnr\<^sub>\<epsilon>_eq:
	assumes "ide u" and "ide v"
	and "src v = trg g" and "src u = trg f"
	and "\<nu> \<in> hom v (u \<star> f)"
	shows "trnr\<^sub>\<epsilon> u \<nu> = (u \<star> \<epsilon>) \<cdot> \<a>[u, f, g] \<cdot> (\<nu> \<star> g)"
	unfolding trnr\<^sub>\<epsilon>_def
	using assms antipar strict_runit comp_ide_arr hcomp_arr_obj by auto

	lemma trnl\<^sub>\<eta>_eq:
	assumes "ide u" and "ide v"
	and "src f = trg v" and "src g = trg u"
	and "\<mu> \<in> hom (f \<star> v) u"
	shows "trnl\<^sub>\<eta> v \<mu> = (g \<star> \<mu>) \<cdot> \<a>[g, f, v] \<cdot> (\<eta> \<star> v)"
	using assms trnl\<^sub>\<eta>_def antipar strict_lunit comp_arr_dom hcomp_obj_arr by auto

	lemma trnl\<^sub>\<epsilon>_eq:
	assumes "ide u" and "ide v"
	and "src f = trg v" and "src g = trg u"
	and "\<nu> \<in> hom v (g \<star> u)"
	shows "trnl\<^sub>\<epsilon> u \<nu> = (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> \<nu>)"
	using assms trnl\<^sub>\<epsilon>_def antipar strict_lunit comp_cod_arr hcomp_obj_arr by auto

	end

	locale adjunction_in_strict_bicategory =
	strict_bicategory +
	adjunction_in_normal_bicategory
	begin

	lemma triangle_left:
	shows "(\<epsilon> \<star> f) \<cdot> (f \<star> \<eta>) = f"
	using ide_left ide_right antipar triangle_left strict_assoc' comp_cod_arr
	by (metis dom_eqI ideD(1) seqE)

	lemma triangle_right:
	shows "(g \<star> \<epsilon>) \<cdot> (\<eta> \<star> g) = g"
	using ide_left ide_right antipar triangle_right strict_assoc comp_cod_arr
	by (metis ideD(1) ideD(2) seqE)

	lemma trnr\<^sub>\<eta>_eq:
	assumes "ide u" and "ide v"
	and "src v = trg g" and "src u = trg f"
	and "\<mu> \<in> hom (v \<star> g) u"
	shows "trnr\<^sub>\<eta> v \<mu> = (\<mu> \<star> f) \<cdot> (v \<star> \<eta>)"
	using assms antipar trnr\<^sub>\<eta>_eq strict_assoc' comp_ide_arr [of "\<a>\<^sup>-\<^sup>1[v, g, f]" "v \<star> \<eta>"]
	by force

	lemma trnr\<^sub>\<epsilon>_eq:
	assumes "ide u" and "ide v"
	and "src v = trg g" and "src u = trg f"
	and "\<nu> \<in> hom v (u \<star> f)"
	shows "trnr\<^sub>\<epsilon> u \<nu> = (u \<star> \<epsilon>) \<cdot> (\<nu> \<star> g)"
	using assms antipar trnr\<^sub>\<epsilon>_eq strict_assoc comp_ide_arr [of "\<a>[u, f, g]" "\<nu> \<star> g"]
	by force

	lemma trnl\<^sub>\<eta>_eq:
	assumes "ide u" and "ide v"
	and "src f = trg v" and "src g = trg u"
	and "\<mu> \<in> hom (f \<star> v) u"
	shows "trnl\<^sub>\<eta> v \<mu> = (g \<star> \<mu>) \<cdot> (\<eta> \<star> v)"
	using assms antipar trnl\<^sub>\<eta>_eq strict_assoc comp_ide_arr [of "\<a>[g, f, v]" "\<eta> \<star> v"]
	by force

	lemma trnl\<^sub>\<epsilon>_eq:
	assumes "ide u" and "ide v"
	and "src f = trg v" and "src g = trg u"
	and "\<nu> \<in> hom v (g \<star> u)"
	shows "trnl\<^sub>\<epsilon> u \<nu> = (\<epsilon> \<star> u) \<cdot> (f \<star> \<nu>)"
	using assms antipar trnl\<^sub>\<epsilon>_eq strict_assoc' comp_ide_arr [of "\<a>\<^sup>-\<^sup>1[f, g, u]" "f \<star> \<nu>"]
	by fastforce

	end

	subsection "Preservation Properties for Adjunctions"

	text \<open>
	Here we show that adjunctions are preserved under isomorphisms of the
	left and right adjoints.
	\<close>

	context bicategory
	begin

	interpretation E: self_evaluation_map V H \<a> \<i> src trg ..
	notation E.eval ("\<lbrace>_\<rbrace>")

	definition adjoint_pair
	where "adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"

	(* These would normally be called "maps", but that name is too heavily used already. *)
	abbreviation is_left_adjoint
	where "is_left_adjoint f \<equiv> \<exists>g. adjoint_pair f g"

	abbreviation is_right_adjoint
	where "is_right_adjoint g \<equiv> \<exists>f. adjoint_pair f g"

	lemma adjoint_pair_antipar:
	assumes "adjoint_pair f g"
	shows "ide f" and "ide g" and "src f = trg g" and "src g = trg f"
	proof -
	obtain \<eta> \<epsilon> where A: "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms adjoint_pair_def by auto
	interpret A: adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using A by auto
	show "ide f" by simp
	show "ide g" by simp
	show "src f = trg g" using A.antipar by simp
	show "src g = trg f" using A.antipar by simp
	qed

	lemma left_adjoint_is_ide:
	assumes "is_left_adjoint f"
	shows "ide f"
	using assms adjoint_pair_antipar by auto

	lemma right_adjoint_is_ide:
	assumes "is_right_adjoint f"
	shows "ide f"
	using assms adjoint_pair_antipar by auto

	lemma adjunction_preserved_by_iso_right:
	assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	and "\<guillemotleft>\<phi> : g \<Rightarrow> g'\<guillemotright>" and "iso \<phi>"
	shows "adjunction_in_bicategory V H \<a> \<i> src trg f g' ((\<phi> \<star> f) \<cdot> \<eta>) (\<epsilon> \<cdot> (f \<star> inv \<phi>))"
	proof
	interpret A: adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using assms by auto
	show "ide f" by simp
	show "ide g'"
	using assms(2) isomorphic_def by auto
	show "\<guillemotleft>(\<phi> \<star> f) \<cdot> \<eta> : src f \<Rightarrow> g' \<star> f\<guillemotright>"
	using assms A.antipar by fastforce
	show "\<guillemotleft>\<epsilon> \<cdot> (f \<star> inv \<phi>) : f \<star> g' \<Rightarrow> src g'\<guillemotright>"
	using assms A.antipar A.counit_in_hom by auto
	show "(\<epsilon> \<cdot> (f \<star> inv \<phi>) \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g', f] \<cdot> (f \<star> (\<phi> \<star> f) \<cdot> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
	proof -
	have "(\<epsilon> \<cdot> (f \<star> inv \<phi>) \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g', f] \<cdot> (f \<star> (\<phi> \<star> f) \<cdot> \<eta>) =
	(\<epsilon> \<star> f) \<cdot> (((f \<star> inv \<phi>) \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g', f]) \<cdot> (f \<star> \<phi> \<star> f) \<cdot> (f \<star> \<eta>)"
	using assms A.antipar whisker_right whisker_left comp_assoc by auto
	also have "... = (\<epsilon> \<star> f) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> inv \<phi> \<star> f)) \<cdot> (f \<star> \<phi> \<star> f) \<cdot> (f \<star> \<eta>)"
	using assms A.antipar assoc'_naturality [of f "inv \<phi>" f] by auto
	also have "... = (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> ((f \<star> inv \<phi> \<star> f) \<cdot> (f \<star> \<phi> \<star> f)) \<cdot> (f \<star> \<eta>)"
	using comp_assoc by simp
	also have "... = (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> g \<star> f) \<cdot> (f \<star> \<eta>)"
	using assms A.antipar comp_inv_arr inv_is_inverse whisker_left
	whisker_right [of f "inv \<phi>" \<phi>]
	by auto
	also have "... = (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>)"
	using assms A.antipar comp_cod_arr by simp
	also have "... = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
	using A.triangle_left by simp
	finally show ?thesis by simp
	qed
	show "(g' \<star> \<epsilon> \<cdot> (f \<star> inv \<phi>)) \<cdot> \<a>[g', f, g'] \<cdot> ((\<phi> \<star> f) \<cdot> \<eta> \<star> g') = \<r>\<^sup>-\<^sup>1[g'] \<cdot> \<l>[g']"
	proof -
	have "(g' \<star> \<epsilon> \<cdot> (f \<star> inv \<phi>)) \<cdot> \<a>[g', f, g'] \<cdot> ((\<phi> \<star> f) \<cdot> \<eta> \<star> g') =
	(g' \<star> \<epsilon>) \<cdot> ((g' \<star> f \<star> inv \<phi>) \<cdot> \<a>[g', f, g']) \<cdot> ((\<phi> \<star> f) \<star> g') \<cdot> (\<eta> \<star> g')"
	using assms A.antipar whisker_left whisker_right comp_assoc by auto
	also have "... = (g' \<star> \<epsilon>) \<cdot> (\<a>[g', f, g] \<cdot> ((g' \<star> f) \<star> inv \<phi>)) \<cdot> ((\<phi> \<star> f) \<star> g') \<cdot> (\<eta> \<star> g')"
	using assms A.antipar assoc_naturality [of g' f "inv \<phi>"] by auto
	also have "... = (g' \<star> \<epsilon>) \<cdot> \<a>[g', f, g] \<cdot> (((g' \<star> f) \<star> inv \<phi>) \<cdot> ((\<phi> \<star> f) \<star> g')) \<cdot> (\<eta> \<star> g')"
	using comp_assoc by simp
	also have "... = (g' \<star> \<epsilon>) \<cdot> (\<a>[g', f, g] \<cdot> ((\<phi> \<star> f) \<star> g)) \<cdot> ((g \<star> f) \<star> inv \<phi>) \<cdot> (\<eta> \<star> g')"
	proof -
	have "((g' \<star> f) \<star> inv \<phi>) \<cdot> ((\<phi> \<star> f) \<star> g') = (\<phi> \<star> f) \<star> inv \<phi>"
	using assms A.antipar comp_arr_dom comp_cod_arr
	interchange [of "g' \<star> f" "\<phi> \<star> f" "inv \<phi>" g']
	by auto
	also have "... = ((\<phi> \<star> f) \<star> g) \<cdot> ((g \<star> f) \<star> inv \<phi>)"
	using assms A.antipar comp_arr_dom comp_cod_arr
	interchange [of "\<phi> \<star> f" "g \<star> f" g "inv \<phi>"]
	by auto
	finally show ?thesis
	using comp_assoc by simp
	qed
	also have "... = ((g' \<star> \<epsilon>) \<cdot> (\<phi> \<star> f \<star> g)) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<cdot> (trg g \<star> inv \<phi>)"
	proof -
	have "\<a>[g', f, g] \<cdot> ((\<phi> \<star> f) \<star> g) = (\<phi> \<star> f \<star> g) \<cdot> \<a>[g, f, g]"
	using assms A.antipar assoc_naturality [of \<phi> f g] by auto
	moreover have "((g \<star> f) \<star> inv \<phi>) \<cdot> (\<eta> \<star> g') = (\<eta> \<star> g) \<cdot> (trg g \<star> inv \<phi>)"
	using assms A.antipar comp_arr_dom comp_cod_arr
	interchange [of "g \<star> f" \<eta> "inv \<phi>" g'] interchange [of \<eta> "trg g" g "inv \<phi>"]
	by auto
	ultimately show ?thesis
	using comp_assoc by simp
	qed
	also have "... = ((\<phi> \<star> src g) \<cdot> (g \<star> \<epsilon>)) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<cdot> (trg g \<star> inv \<phi>)"
	using assms A.antipar comp_arr_dom comp_cod_arr
	interchange [of g' \<phi> \<epsilon> "f \<star> g"] interchange [of \<phi> g "src g" \<epsilon>]
	by (metis A.counit_simps(1) A.counit_simps(2) A.counit_simps(3) in_homE)
	also have "... = (\<phi> \<star> src g) \<cdot> ((g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g)) \<cdot> (trg g \<star> inv \<phi>)"
	using comp_assoc by simp
	also have "... = ((\<phi> \<star> src g) \<cdot> \<r>\<^sup>-\<^sup>1[g]) \<cdot> \<l>[g] \<cdot> (trg g \<star> inv \<phi>)"
	using assms A.antipar A.triangle_right comp_cod_arr comp_assoc
	by simp
	also have "... = (\<r>\<^sup>-\<^sup>1[g'] \<cdot> \<phi>) \<cdot> inv \<phi> \<cdot> \<l>[g']"
	using assms A.antipar runit'_naturality [of \<phi>] lunit_naturality [of "inv \<phi>"]
	by auto
	also have "... = \<r>\<^sup>-\<^sup>1[g'] \<cdot> (\<phi> \<cdot> inv \<phi>) \<cdot> \<l>[g']"
	using comp_assoc by simp
	also have "... = \<r>\<^sup>-\<^sup>1[g'] \<cdot> \<l>[g']"
	using assms comp_cod_arr comp_arr_inv' by auto
	finally show ?thesis by simp
	qed
	qed

	lemma adjunction_preserved_by_iso_left:
	assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>" and "iso \<phi>"
	shows "adjunction_in_bicategory V H \<a> \<i> src trg f' g ((g \<star> \<phi>) \<cdot> \<eta>) (\<epsilon> \<cdot> (inv \<phi> \<star> g))"
	proof
	interpret A: adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using assms by auto
	show "ide g" by simp
	show "ide f'"
	using assms(2) isomorphic_def by auto
	show "\<guillemotleft>(g \<star> \<phi>) \<cdot> \<eta> : src f' \<Rightarrow> g \<star> f'\<guillemotright>"
	using assms A.antipar A.unit_in_hom by force
	show "\<guillemotleft>\<epsilon> \<cdot> (inv \<phi> \<star> g) : f' \<star> g \<Rightarrow> src g\<guillemotright>"
	using assms A.antipar by force
	show "(g \<star> \<epsilon> \<cdot> (inv \<phi> \<star> g)) \<cdot> \<a>[g, f', g] \<cdot> ((g \<star> \<phi>) \<cdot> \<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
	proof -
	have "(g \<star> \<epsilon> \<cdot> (inv \<phi> \<star> g)) \<cdot> \<a>[g, f', g] \<cdot> ((g \<star> \<phi>) \<cdot> \<eta> \<star> g) =
	(g \<star> \<epsilon>) \<cdot> ((g \<star> inv \<phi> \<star> g) \<cdot> \<a>[g, f', g]) \<cdot> ((g \<star> \<phi>) \<star> g) \<cdot> (\<eta> \<star> g)"
	using assms A.antipar whisker_left whisker_right comp_assoc by auto
	also have "... = (g \<star> \<epsilon>) \<cdot> (\<a>[g, f, g] \<cdot> ((g \<star> inv \<phi>) \<star> g)) \<cdot> ((g \<star> \<phi>) \<star> g) \<cdot> (\<eta> \<star> g)"
	using assms A.antipar assoc_naturality [of g "inv \<phi>" g] by auto
	also have "... = (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (((g \<star> inv \<phi>) \<star> g) \<cdot> ((g \<star> \<phi>) \<star> g)) \<cdot> (\<eta> \<star> g)"
	using comp_assoc by simp
	also have "... = (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> ((g \<star> f) \<star> g) \<cdot> (\<eta> \<star> g)"
	using assms A.antipar comp_inv_arr inv_is_inverse whisker_right
	whisker_left [of g "inv \<phi>" \<phi>]
	by auto
	also have "... = (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g)"
	using assms A.antipar comp_cod_arr by simp
	also have "... = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
	using A.triangle_right by simp
	finally show ?thesis by simp
	qed
	show "(\<epsilon> \<cdot> (inv \<phi> \<star> g) \<star> f') \<cdot> \<a>\<^sup>-\<^sup>1[f', g, f'] \<cdot> (f' \<star> (g \<star> \<phi>) \<cdot> \<eta>) = \<l>\<^sup>-\<^sup>1[f'] \<cdot> \<r>[f']"
	proof -
	have "(\<epsilon> \<cdot> (inv \<phi> \<star> g) \<star> f') \<cdot> \<a>\<^sup>-\<^sup>1[f', g, f'] \<cdot> (f' \<star> (g \<star> \<phi>) \<cdot> \<eta>) =
	(\<epsilon> \<star> f') \<cdot> (((inv \<phi> \<star> g) \<star> f') \<cdot> \<a>\<^sup>-\<^sup>1[f', g, f']) \<cdot> (f' \<star> g \<star> \<phi>) \<cdot> (f' \<star> \<eta>)"
	using assms A.antipar whisker_right whisker_left comp_assoc
	by auto
	also have "... = (\<epsilon> \<star> f') \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f'] \<cdot> (inv \<phi> \<star> g \<star> f')) \<cdot> (f' \<star> g \<star> \<phi>) \<cdot> (f' \<star> \<eta>)"
	using assms A.antipar assoc'_naturality [of "inv \<phi>" g f'] by auto
	also have "... = (\<epsilon> \<star> f') \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f'] \<cdot> ((inv \<phi> \<star> g \<star> f') \<cdot> (f' \<star> g \<star> \<phi>)) \<cdot> (f' \<star> \<eta>)"
	using comp_assoc by simp
	also have "... = (\<epsilon> \<star> f') \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, f'] \<cdot> (f \<star> g \<star> \<phi>)) \<cdot> (inv \<phi> \<star> g \<star> f) \<cdot> (f' \<star> \<eta>)"
	proof -
	have "(inv \<phi> \<star> g \<star> f') \<cdot> (f' \<star> g \<star> \<phi>) = (f \<star> g \<star> \<phi>) \<cdot> (inv \<phi> \<star> g \<star> f)"
	using assms(2-3) A.antipar comp_arr_dom comp_cod_arr
	interchange [of "inv \<phi>" f' "g \<star> f'" "g \<star> \<phi>"]
	interchange [of f "inv \<phi>" "g \<star> \<phi>" "g \<star> f"]
	by auto
	thus ?thesis
	using comp_assoc by simp
	qed
	also have "... = ((\<epsilon> \<star> f') \<cdot> ((f \<star> g) \<star> \<phi>)) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<cdot> (inv \<phi> \<star> src f)"
	proof -
	have "\<a>\<^sup>-\<^sup>1[f, g, f'] \<cdot> (f \<star> g \<star> \<phi>) = ((f \<star> g) \<star> \<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f]"
	using assms A.antipar assoc'_naturality [of f g \<phi>] by auto
	moreover have "(inv \<phi> \<star> g \<star> f) \<cdot> (f' \<star> \<eta>) = (f \<star> \<eta>) \<cdot> (inv \<phi> \<star> src f)"
	using assms A.antipar comp_arr_dom comp_cod_arr
	interchange [of "inv \<phi>" f' "g \<star> f" \<eta>] interchange [of f "inv \<phi>" \<eta> "src f"]
	by auto
	ultimately show ?thesis
	using comp_assoc by simp
	qed
	also have "... = ((trg f \<star> \<phi>) \<cdot> (\<epsilon> \<star> f)) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<cdot> (inv \<phi> \<star> src f)"
	using assms A.antipar comp_arr_dom comp_cod_arr
	interchange [of \<epsilon> "f \<star> g" f' \<phi>] interchange [of "trg f" \<epsilon> \<phi> f]
	by (metis A.counit_simps(1) A.counit_simps(2) A.counit_simps(3) in_homE)
	also have "... = (trg f \<star> \<phi>) \<cdot> ((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>)) \<cdot> (inv \<phi> \<star> src f)"
	using comp_assoc by simp
	also have "... = ((trg f \<star> \<phi>) \<cdot> \<l>\<^sup>-\<^sup>1[f]) \<cdot> \<r>[f] \<cdot> (inv \<phi> \<star> src f)"
	using assms A.antipar A.triangle_left comp_cod_arr comp_assoc
	by simp
	also have "... = (\<l>\<^sup>-\<^sup>1[f'] \<cdot> \<phi>) \<cdot> inv \<phi> \<cdot> \<r>[f']"
	using assms A.antipar lunit'_naturality runit_naturality [of "inv \<phi>"] by auto
	also have "... = \<l>\<^sup>-\<^sup>1[f'] \<cdot> (\<phi> \<cdot> inv \<phi>) \<cdot> \<r>[f']"
	using comp_assoc by simp
	also have "... = \<l>\<^sup>-\<^sup>1[f'] \<cdot> \<r>[f']"
	using assms comp_cod_arr comp_arr_inv inv_is_inverse by auto
	finally show ?thesis by simp
	qed
	qed

	lemma adjoint_pair_preserved_by_iso:
	assumes "adjoint_pair f g"
	and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>" and "iso \<phi>"
	and "\<guillemotleft>\<psi> : g \<Rightarrow> g'\<guillemotright>" and "iso \<psi>"
	shows "adjoint_pair f' g'"
	using assms adjoint_pair_def adjunction_preserved_by_iso_left
	adjunction_preserved_by_iso_right
	by metis

	lemma left_adjoint_preserved_by_iso:
	assumes "is_left_adjoint f"
	and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>" and "iso \<phi>"
	shows "is_left_adjoint f'"
	proof -
	obtain g where g: "adjoint_pair f g"
	using assms by auto
	have "adjoint_pair f' g"
	using assms g adjoint_pair_preserved_by_iso [of f g \<phi> f' g g]
	adjoint_pair_antipar [of f g]
	by auto
	thus ?thesis by auto
	qed

	lemma right_adjoint_preserved_by_iso:
	assumes "is_right_adjoint g"
	and "\<guillemotleft>\<phi> : g \<Rightarrow> g'\<guillemotright>" and "iso \<phi>"
	shows "is_right_adjoint g'"
	proof -
	obtain f where f: "adjoint_pair f g"
	using assms by auto
	have "adjoint_pair f g'"
	using assms f adjoint_pair_preserved_by_iso [of f g f f \<phi> g']
	adjoint_pair_antipar [of f g]
	by auto
	thus ?thesis by auto
	qed

	lemma left_adjoint_preserved_by_iso':
	assumes "is_left_adjoint f" and "f \<cong> f'"
	shows "is_left_adjoint f'"
	using assms isomorphic_def left_adjoint_preserved_by_iso by blast

	lemma right_adjoint_preserved_by_iso':
	assumes "is_right_adjoint g" and "g \<cong> g'"
	shows "is_right_adjoint g'"
	using assms isomorphic_def right_adjoint_preserved_by_iso by blast

	lemma obj_self_adjunction:
	assumes "obj a"
	shows "adjunction_in_bicategory V H \<a> \<i> src trg a a \<l>\<^sup>-\<^sup>1[a] \<r>[a]"
	proof
	show 1: "ide a"
	using assms by auto
	show "\<guillemotleft>\<l>\<^sup>-\<^sup>1[a] : src a \<Rightarrow> a \<star> a\<guillemotright>"
	using assms 1 by auto
	show "\<guillemotleft>\<r>[a] : a \<star> a \<Rightarrow> src a\<guillemotright>"
	using assms 1 by fastforce
	show "(\<r>[a] \<star> a) \<cdot> \<a>\<^sup>-\<^sup>1[a, a, a] \<cdot> (a \<star> \<l>\<^sup>-\<^sup>1[a]) = \<l>\<^sup>-\<^sup>1[a] \<cdot> \<r>[a]"
	using assms 1 canI_unitor_1 canI_associator_1(2) canI_associator_3
	whisker_can_right_1 whisker_can_left_1 can_Ide_self obj_simps
	by simp
	show "(a \<star> \<r>[a]) \<cdot> \<a>[a, a, a] \<cdot> (\<l>\<^sup>-\<^sup>1[a] \<star> a) = \<r>\<^sup>-\<^sup>1[a] \<cdot> \<l>[a]"
	using assms 1 canI_unitor_1 canI_associator_1(2) canI_associator_3
	whisker_can_right_1 whisker_can_left_1 can_Ide_self
	by simp
	qed

	lemma obj_is_self_adjoint:
	assumes "obj a"
	shows "adjoint_pair a a" and "is_left_adjoint a" and "is_right_adjoint a"
	using assms obj_self_adjunction adjoint_pair_def by auto

	end

	subsection "Pseudofunctors and Adjunctions"

	context pseudofunctor
	begin

	lemma preserves_adjunction:
	assumes "adjunction_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_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_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
	interpret A: adjunction_data_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 adjunction_data_in_bicategory_axioms preserves_adjunction_data by auto
	show "adjunction_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
	show "(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D
	(F f \<star>\<^sub>D D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) =
	D.lunit' (F f) \<cdot>\<^sub>D \<r>\<^sub>D[F f]"
	proof -
	have 1: "D.iso (\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f)))"
	using antipar C.VV.ide_char C.VV.arr_char D.iso_is_arr FF_def
	by (intro D.isos_compose D.seqI, simp_all)
	have "(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D
	(F f \<star>\<^sub>D 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) \<star>\<^sub>D F f) \<cdot>\<^sub>D
	(D.inv (\<Phi> (f, g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D
	\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f)) \<cdot>\<^sub>D
	(F f \<star>\<^sub>D D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))"
	proof -
	have "\<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] =
	(D.inv (\<Phi> (f, g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D
	\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f))"
	proof -
	have "\<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D D.inv (\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f))) =
	D.inv (F \<a>\<^sub>C[f, g, f] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f))"
	proof -
	have "D.inv (\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F f]) =
	D.inv (F \<a>\<^sub>C[f, g, f] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f))"
	using antipar assoc_coherence by simp
	moreover
	have "D.inv (\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F f]) =
	\<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D D.inv (\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f)))"
	proof -
	have "D.seq (\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f))) \<a>\<^sub>D[F f, F g, F f]"
	using antipar by fastforce
	thus ?thesis
	using 1 antipar D.comp_assoc
	D.inv_comp [of "\<a>\<^sub>D[F f, F g, F f]" "\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f))"]
	by auto
	qed
	ultimately show ?thesis by simp
	qed
	moreover have 2: "D.iso (F \<a>\<^sub>C[f, g, f] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f))"
	using antipar D.isos_compose C.VV.ide_char C.VV.arr_char cmp_simps(4)
	by simp
	ultimately have "\<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] =
	D.inv (F \<a>\<^sub>C[f, g, f] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f)) \<cdot>\<^sub>D
	(\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f)))"
	using 1 2 antipar D.invert_side_of_triangle(2) D.inv_inv D.iso_inv_iso D.arr_inv
	by metis
	moreover have "D.inv (F \<a>\<^sub>C[f, g, f] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f)) =
	(D.inv (\<Phi> (f, g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f]"
	proof -
	have "D.inv (F \<a>\<^sub>C[f, g, f] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f)) =
	D.inv (\<Phi> (f \<star>\<^sub>C g, f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f)) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f]"
	using antipar D.isos_compose C.VV.arr_char cmp_simps(4)
	preserves_inv D.inv_comp C.VV.cod_char
	by simp
	also have "... = (D.inv (\<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, f))) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f]"
	using antipar D.inv_comp C.VV.ide_char C.VV.arr_char cmp_simps(4)
	by simp
	also have "... = ((D.inv (\<Phi> (f, g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, f))) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f]"
	using antipar C.VV.ide_char C.VV.arr_char by simp
	also have "... = (D.inv (\<Phi> (f, g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f]"
	using D.comp_assoc by simp
	finally show ?thesis by simp
	qed
	ultimately show ?thesis
	using D.comp_assoc by simp
	qed
	thus ?thesis
	using D.comp_assoc by simp
	qed
	also have "... = (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<star>\<^sub>D F f) \<cdot>\<^sub>D
	D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D
	\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D
	(F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))"
	proof -
	have "... = ((D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f)) \<cdot>\<^sub>D
	(D.inv (\<Phi> (f, g)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D
	\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, f)) \<cdot>\<^sub>D
	((F f \<star>\<^sub>D D.inv (\<Phi> (g, f))) \<cdot>\<^sub>D (F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)))"
	proof -
	have "D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) \<star>\<^sub>D F f =
	(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f)"
	using ide_left ide_right antipar D.whisker_right unit_char(2)
	by (metis A.counit_simps(1) A.ide_left D.comp_assoc)
	moreover have "F f \<star>\<^sub>D D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) =
	(F f \<star>\<^sub>D D.inv (\<Phi> (g, f))) \<cdot>\<^sub>D (F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))"
	using antipar unit_char(2) D.whisker_left by simp
	ultimately show ?thesis by simp
	qed
	also have "... = (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<star>\<^sub>D F f) \<cdot>\<^sub>D
	(((\<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D (D.inv (\<Phi> (f, g)) \<star>\<^sub>D F f)) \<cdot>\<^sub>D
	D.inv (\<Phi> (f \<star>\<^sub>C g, f))) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D \<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D
	(((F f \<star>\<^sub>D \<Phi> (g, f)) \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (\<Phi> (g, f)))) \<cdot>\<^sub>D
	(F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)))"
	using D.comp_assoc by simp
	also have "... = (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<star>\<^sub>D F f) \<cdot>\<^sub>D
	D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D
	\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D
	(F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))"
	proof -
	have "((F f \<star>\<^sub>D \<Phi> (g, f)) \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (\<Phi> (g, f)))) \<cdot>\<^sub>D
	(F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) =
	F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)"
	proof -
	have "(F f \<star>\<^sub>D \<Phi> (g, f)) \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (\<Phi> (g, f))) = F f \<star>\<^sub>D F (g \<star>\<^sub>C f)"
	using antipar unit_char(2) D.comp_arr_inv D.inv_is_inverse
	D.whisker_left [of "F f" "\<Phi> (g, f)" "D.inv (\<Phi> (g, f))"]
	by simp
	moreover have "D.seq (F f \<star>\<^sub>D F (g \<star>\<^sub>C f)) (F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))"
	using antipar by fastforce
	ultimately show ?thesis
	using D.comp_cod_arr by auto
	qed
	moreover have "((\<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D (D.inv (\<Phi> (f, g)) \<star>\<^sub>D F f)) \<cdot>\<^sub>D
	D.inv (\<Phi> (f \<star>\<^sub>C g, f)) =
	D.inv (\<Phi> (f \<star>\<^sub>C g, f))"
	using antipar D.comp_arr_inv D.inv_is_inverse D.comp_cod_arr
	D.whisker_right [of "F f" "\<Phi> (f, g)" "D.inv (\<Phi> (f, g))"]
	by simp
	ultimately show ?thesis by simp
	qed
	finally show ?thesis by simp
	qed
	also have "... = (D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D
	D.inv (\<Phi> (trg\<^sub>C f, f)) \<cdot>\<^sub>D F (\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>D
	((\<Phi> (f \<star>\<^sub>C g, f) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, f))) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f]) \<cdot>\<^sub>D
	((\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D D.inv (\<Phi> (f, g \<star>\<^sub>C f))) \<cdot>\<^sub>D F (f \<star>\<^sub>C \<eta>)) \<cdot>\<^sub>D
	\<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))"
	proof -
	have "(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<star>\<^sub>D F f) \<cdot>\<^sub>D
	D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D
	\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D
	(F f \<star>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) =
	((D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D (F \<epsilon> \<star>\<^sub>D F f)) \<cdot>\<^sub>D
	D.inv (\<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D
	F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D
	\<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D
	((F f \<star>\<^sub>D F \<eta>) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f)))"
	using antipar D.comp_assoc D.whisker_left D.whisker_right unit_char(2)
	by simp
	moreover have "F \<epsilon> \<star>\<^sub>D F f = D.inv (\<Phi> (trg\<^sub>C f, f)) \<cdot>\<^sub>D F (\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f)"
	using antipar \<Phi>.naturality [of "(\<epsilon>, f)"] C.VV.arr_char FF_def
	D.invert_side_of_triangle(1) C.VV.dom_char C.VV.cod_char
	by simp
	moreover have "F f \<star>\<^sub>D F \<eta> = D.inv (\<Phi> (f, g \<star>\<^sub>C f)) \<cdot>\<^sub>D F (f \<star>\<^sub>C \<eta>) \<cdot>\<^sub>D \<Phi> (f, src\<^sub>C f)"
	using antipar \<Phi>.naturality [of "(f, \<eta>)"] C.VV.arr_char FF_def
	D.invert_side_of_triangle(1) C.VV.dom_char C.VV.cod_char
	by simp
	ultimately show ?thesis
	using D.comp_assoc by simp
	qed
	also have "... = ((D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f))) \<cdot>\<^sub>D
	(F (\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>D
	(F ((f \<star>\<^sub>C g) \<star>\<^sub>C f) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D F (f \<star>\<^sub>C g \<star>\<^sub>C f)) \<cdot>\<^sub>D
	F (f \<star>\<^sub>C \<eta>)) \<cdot>\<^sub>D
	\<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))"
	using antipar D.comp_arr_inv' D.comp_assoc by simp
	also have "... = ((D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f))) \<cdot>\<^sub>D
	(F (\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D F (f \<star>\<^sub>C \<eta>)) \<cdot>\<^sub>D
	\<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))"
	proof -
	have "F ((f \<star>\<^sub>C g) \<star>\<^sub>C f) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D F (f \<star>\<^sub>C g \<star>\<^sub>C f) = F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f]"
	using antipar D.comp_arr_dom D.comp_cod_arr by simp
	thus ?thesis by simp
	qed
	also have "... = D.inv (\<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)) \<cdot>\<^sub>D
	F ((\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>)) \<cdot>\<^sub>D
	\<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))"
	proof -
	have "(D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f)) =
	D.inv (\<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f))"
	proof -
	have "D.iso (\<Phi> (trg\<^sub>C f, f))"
	using antipar by simp
	moreover have "D.iso (unit (trg\<^sub>C f) \<star>\<^sub>D F f)"
	using antipar unit_char(2) by simp
	moreover have "D.seq (\<Phi> (trg\<^sub>C f, f)) (unit (trg\<^sub>C f) \<star>\<^sub>D F f)"
	using antipar D.iso_is_arr calculation(2)
	apply (intro D.seqI D.hseqI) by auto
	ultimately show ?thesis
	using antipar D.inv_comp unit_char(2) by simp
	qed
	moreover have "F (\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D F (f \<star>\<^sub>C \<eta>) =
	F ((\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>))"
	using antipar by simp
	ultimately show ?thesis by simp
	qed
	also have "... = (D.lunit' (F f) \<cdot>\<^sub>D F \<l>\<^sub>C[f]) \<cdot>\<^sub>D
	F (C.lunit' f \<cdot>\<^sub>C \<r>\<^sub>C[f]) \<cdot>\<^sub>D
	(D.inv (F \<r>\<^sub>C[f]) \<cdot>\<^sub>D \<r>\<^sub>D[F f])"
	proof -
	have "F ((\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>)) = F (C.lunit' f \<cdot>\<^sub>C \<r>\<^sub>C[f])"
	using triangle_left by simp
	moreover have "D.inv (\<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)) =
	D.lunit' (F f) \<cdot>\<^sub>D F \<l>\<^sub>C[f]"
	proof -
	have 0: "D.iso (\<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f))"
	using unit_char(2)
	apply (intro D.isos_compose D.seqI) by auto
	show ?thesis
	proof -
	have 1: "D.iso (F \<l>\<^sub>C[f])"
	using C.iso_lunit preserves_iso by auto
	moreover have "D.iso (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 (no_types) A.ide_left D.iso_lunit ide_left lunit_coherence)
	moreover have "D.inv (D.inv (F \<l>\<^sub>C[f])) = F \<l>\<^sub>C[f]"
	using 1 D.inv_inv by blast
	ultimately show ?thesis
	by (metis 0 D.inv_comp D.invert_side_of_triangle(2) D.iso_inv_iso
	D.iso_is_arr ide_left lunit_coherence)
	qed
	qed
	moreover have "\<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f)) = D.inv (F \<r>\<^sub>C[f]) \<cdot>\<^sub>D \<r>\<^sub>D[F f]"
	using ide_left runit_coherence preserves_iso C.iso_runit D.invert_side_of_triangle(1)
	by (metis A.ide_left D.runit_simps(1))
	ultimately show ?thesis by simp
	qed
	also have "... = D.lunit' (F f) \<cdot>\<^sub>D
	((F \<l>\<^sub>C[f] \<cdot>\<^sub>D F (C.lunit' f)) \<cdot>\<^sub>D (F \<r>\<^sub>C[f] \<cdot>\<^sub>D D.inv (F \<r>\<^sub>C[f]))) \<cdot>\<^sub>D
	\<r>\<^sub>D[F f]"
	using D.comp_assoc by simp
	also have "... = D.lunit' (F f) \<cdot>\<^sub>D \<r>\<^sub>D[F f]"
	using D.comp_cod_arr C.iso_runit C.iso_lunit preserves_iso D.comp_arr_inv'
	preserves_inv
	by force
	finally show ?thesis by blast
	qed
	show "(F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D
	\<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g) =
	D.runit' (F g) \<cdot>\<^sub>D \<l>\<^sub>D[F g]"
	proof -
	have "\<a>\<^sub>D[F g, F f, F g] =
	D.inv (\<Phi> (g, f \<star>\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g))) \<cdot>\<^sub>D
	F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D \<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D (\<Phi> (g, f) \<star>\<^sub>D F g)"
	proof -
	have "D.iso (\<Phi> (g, f \<star>\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g)))"
	using antipar D.iso_is_arr
	apply (intro D.isos_compose, auto)
	by (metis C.iso_assoc D.comp_assoc D.seqE ide_left ide_right
	preserves_assoc(1) preserves_iso)
	moreover have "F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D \<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D (\<Phi> (g, f) \<star>\<^sub>D F g) =
	\<Phi> (g, f \<star>\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D \<a>\<^sub>D[F g, F f, F g]"
	using antipar assoc_coherence by simp
	moreover have "D.seq (F \<a>\<^sub>C[g, f, g]) (\<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D (\<Phi> (g, f) \<star>\<^sub>D F g))"
	using antipar C.VV.arr_char C.VV.dom_char C.VV.cod_char FF_def
	by (intro D.seqI D.hseqI') auto
	ultimately show ?thesis
	using D.invert_side_of_triangle(1) D.comp_assoc by auto
	qed
	hence "(F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D
	\<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D
	(D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g) =
	(F g \<star>\<^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> (g, f \<star>\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g))) \<cdot>\<^sub>D
	F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D
	\<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D (\<Phi> (g, f) \<star>\<^sub>D F g) \<cdot>\<^sub>D
	(D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g)"
	using D.comp_assoc by simp
	also have "... = ((F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g))) \<cdot>\<^sub>D
	D.inv (\<Phi> (g, f \<star>\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g))) \<cdot>\<^sub>D
	F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D \<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D
	(\<Phi> (g, f) \<star>\<^sub>D F g) \<cdot>\<^sub>D ((D.inv (\<Phi> (g, f)) \<star>\<^sub>D F g) \<cdot>\<^sub>D
	(F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g))"
	proof -
	have "F g \<star>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D \<Phi> (f, g) =
	(F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g))"
	proof -
	have "D.seq (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon>) (\<Phi> (f, g))"
	using antipar D.comp_assoc by simp
	thus ?thesis
	using antipar D.whisker_left by simp
	qed
	moreover have "D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g =
	(D.inv (\<Phi> (g, f)) \<star>\<^sub>D F g) \<cdot>\<^sub>D (F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g)"
	using antipar D.whisker_right by simp
	ultimately show ?thesis
	using D.comp_assoc by simp
	qed
	also have "... = (F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D
	(((F g \<star>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D D.inv (F g \<star>\<^sub>D \<Phi> (f, g))) \<cdot>\<^sub>D
	D.inv (\<Phi> (g, f \<star>\<^sub>C g))) \<cdot>\<^sub>D F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D \<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D
	((\<Phi> (g, f) \<star>\<^sub>D F g) \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<star>\<^sub>D F g)) \<cdot>\<^sub>D
	(F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g)"
	proof -
	have "D.inv (\<Phi> (g, f \<star>\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g))) =
	D.inv (F g \<star>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D D.inv (\<Phi> (g, f \<star>\<^sub>C g))"
	proof -
	have "D.iso (\<Phi> (g, f \<star>\<^sub>C g))"
	using antipar by simp
	moreover have "D.iso (F g \<star>\<^sub>D \<Phi> (f, g))"
	using antipar by simp
	moreover have "D.seq (\<Phi> (g, f \<star>\<^sub>C g)) (F g \<star>\<^sub>D \<Phi> (f, g))"
	using antipar cmp_in_hom A.ide_right D.iso_is_arr
	by (intro D.seqI) auto
	ultimately show ?thesis
	using antipar D.inv_comp by simp
	qed
	thus ?thesis
	using D.comp_assoc by simp
	qed
	also have "... = (F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D
	D.inv (\<Phi> (g, f \<star>\<^sub>C g)) \<cdot>\<^sub>D F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D \<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D
	(F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g)"
	proof -
	have "((\<Phi> (g, f) \<star>\<^sub>D F g) \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<star>\<^sub>D F g)) \<cdot>\<^sub>D
	(F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g) =
	(F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g)"
	proof -
	have "(\<Phi> (g, f) \<star>\<^sub>D F g) \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<star>\<^sub>D F g) = F (g \<star>\<^sub>C f) \<star>\<^sub>D F g"
	using antipar D.comp_arr_inv'
	D.whisker_right [of "F g" "\<Phi> (g, f)" "D.inv (\<Phi> (g, f))"]
	by simp
	thus ?thesis
	using antipar D.comp_cod_arr D.whisker_right by simp
	qed
	moreover have "((F g \<star>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D D.inv (F g \<star>\<^sub>D \<Phi> (f, g))) \<cdot>\<^sub>D
	D.inv (\<Phi> (g, f \<star>\<^sub>C g)) =
	D.inv (\<Phi> (g, f \<star>\<^sub>C g))"
	using antipar D.comp_arr_inv' D.comp_cod_arr
	D.whisker_left [of "F g" "\<Phi> (f, g)" "D.inv (\<Phi> (f, g))"]
	by simp
	ultimately show ?thesis by simp
	qed
	also have "... = (F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f))) \<cdot>\<^sub>D
	((F g \<star>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D D.inv (\<Phi> (g, f \<star>\<^sub>C g))) \<cdot>\<^sub>D
	F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D
	(\<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D (F \<eta> \<star>\<^sub>D F g)) \<cdot>\<^sub>D
	(unit (src\<^sub>C f) \<star>\<^sub>D F g)"
	using antipar D.whisker_left D.whisker_right unit_char(2) D.comp_assoc by simp
	also have "... = (F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (g, src\<^sub>C g)) \<cdot>\<^sub>D
	(F (g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>D F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D F (\<eta> \<star>\<^sub>C g)) \<cdot>\<^sub>D
	\<Phi> (trg\<^sub>C g, g) \<cdot>\<^sub>D (unit (src\<^sub>C f) \<star>\<^sub>D F g)"
	proof -
	have "(F g \<star>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D D.inv (\<Phi> (g, f \<star>\<^sub>C g)) = D.inv (\<Phi> (g, src\<^sub>C g)) \<cdot>\<^sub>D F (g \<star>\<^sub>C \<epsilon>)"
	using antipar C.VV.arr_char \<Phi>.naturality [of "(g, \<epsilon>)"] FF_def
	D.invert_opposite_sides_of_square C.VV.dom_char C.VV.cod_char
	by simp
	moreover have "\<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D (F \<eta> \<star>\<^sub>D F g) = F (\<eta> \<star>\<^sub>C g) \<cdot>\<^sub>D \<Phi> (trg\<^sub>C g, g)"
	using antipar C.VV.arr_char \<Phi>.naturality [of "(\<eta>, g)"] FF_def
	C.VV.dom_char C.VV.cod_char
	by simp
	ultimately show ?thesis
	using D.comp_assoc by simp
	qed
	also have "... = ((F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (g, src\<^sub>C g)) \<cdot>\<^sub>D
	F (C.runit' g)) \<cdot>\<^sub>D (F \<l>\<^sub>C[g] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C g, g) \<cdot>\<^sub>D (unit (src\<^sub>C f) \<star>\<^sub>D F g))"
	proof -
	have "F (g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>D F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D F (\<eta> \<star>\<^sub>C g) = F (C.runit' g) \<cdot>\<^sub>D F \<l>\<^sub>C[g]"
	using ide_left ide_right antipar triangle_right
	by (metis C.comp_in_homE C.seqI' preserves_comp triangle_in_hom(2))
	thus ?thesis
	using D.comp_assoc by simp
	qed
	also have "... = D.runit' (F g) \<cdot>\<^sub>D \<l>\<^sub>D[F g]"
	proof -
	have "D.inv \<r>\<^sub>D[F g] =
	(F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (g, src\<^sub>C g)) \<cdot>\<^sub>D F (C.runit' g)"
	proof -
	have "D.runit' (F g) = D.inv (F \<r>\<^sub>C[g] \<cdot>\<^sub>D \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (src\<^sub>C g)))"
	using runit_coherence by simp
	also have
	"... = (F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (g, src\<^sub>C g)) \<cdot>\<^sub>D F (C.runit' g)"
	proof -
	have "D.inv (F \<r>\<^sub>C[g] \<cdot>\<^sub>D \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (src\<^sub>C g))) =
	D.inv (F g \<star>\<^sub>D unit (src\<^sub>C g)) \<cdot>\<^sub>D D.inv (\<Phi> (g, src\<^sub>C g)) \<cdot>\<^sub>D F (C.runit' g)"
	proof -
	have "D.iso (F \<r>\<^sub>C[g])"
	using preserves_iso by simp
	moreover have 1: "D.iso (\<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (src\<^sub>C g)))"
	using preserves_iso unit_char(2) D.arrI D.seqE ide_right runit_coherence
	by (intro D.isos_compose D.seqI, auto)
	moreover have "D.seq (F \<r>\<^sub>C[g]) (\<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (src\<^sub>C g)))"
	using ide_right A.ide_right D.runit_simps(1) runit_coherence by metis
	ultimately have "D.inv (F \<r>\<^sub>C[g] \<cdot>\<^sub>D \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (src\<^sub>C g))) =
	D.inv (\<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (src\<^sub>C g))) \<cdot>\<^sub>D F (C.runit' g)"
	using C.iso_runit preserves_inv D.inv_comp by simp
	moreover have "D.inv (\<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (src\<^sub>C g))) =
	D.inv (F g \<star>\<^sub>D unit (src\<^sub>C g)) \<cdot>\<^sub>D D.inv (\<Phi> (g, src\<^sub>C g))"
	proof -
	have "D.seq (\<Phi> (g, src\<^sub>C g)) (F g \<star>\<^sub>D unit (src\<^sub>C g))"
	using 1 antipar preserves_iso unit_char(2) by fast
	(*
	* TODO: The fact that auto cannot do this step is probably what is blocking
	* the whole thing from being done by auto.
	*)
	thus ?thesis
	using 1 antipar preserves_iso unit_char(2) D.inv_comp by auto
	qed
	ultimately show ?thesis
	using D.comp_assoc by simp
	qed
	thus ?thesis
	using antipar unit_char(2) preserves_iso by simp
	qed
	finally show ?thesis by simp
	qed
	thus ?thesis
	using antipar lunit_coherence by simp
	qed
	finally show ?thesis by simp
	qed
	qed
	qed

	lemma preserves_adjoint_pair:
	assumes "C.adjoint_pair f g"
	shows "D.adjoint_pair (F f) (F g)"
	using assms C.adjoint_pair_def D.adjoint_pair_def preserves_adjunction by blast

	lemma preserves_left_adjoint:
	assumes "C.is_left_adjoint f"
	shows "D.is_left_adjoint (F f)"
	using assms preserves_adjoint_pair by auto

	lemma preserves_right_adjoint:
	assumes "C.is_right_adjoint g"
	shows "D.is_right_adjoint (F g)"
	using assms preserves_adjoint_pair by auto

	end

	context equivalence_pseudofunctor
	begin

	lemma reflects_adjunction:
	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 "adjunction_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 "adjunction_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 -
	let ?\<eta>' = "D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)"
	let ?\<epsilon>' = "D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)"
	interpret A': adjunction_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> ?\<eta>' ?\<epsilon>'
	using assms(5) by auto
	interpret A: 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(1-4) by (unfold_locales, auto)
	show ?thesis
	proof
	show "(\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>) = \<l>\<^sub>C\<^sup>-\<^sup>1[f] \<cdot>\<^sub>C \<r>\<^sub>C[f]"
	proof -
	have 1: "C.par ((\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>)) (\<l>\<^sub>C\<^sup>-\<^sup>1[f] \<cdot>\<^sub>C \<r>\<^sub>C[f])"
	using assms A.antipar by simp
	moreover have "F ((\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>)) = F (\<l>\<^sub>C\<^sup>-\<^sup>1[f] \<cdot>\<^sub>C \<r>\<^sub>C[f])"
	proof -
	have "F ((\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>)) =
	F (\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>D F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>D F (f \<star>\<^sub>C \<eta>)"
	using 1 by (metis C.seqE preserves_comp)
	also have "... =
	(F (\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f)) \<cdot>\<^sub>D
	(\<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (\<Phi> (g, f))) \<cdot>\<^sub>D
	(D.inv (\<Phi> (f, g \<star>\<^sub>C f)) \<cdot>\<^sub>D F (f \<star>\<^sub>C \<eta>))"
	using assms A.antipar preserves_assoc(2) D.comp_assoc by auto
	also have "... = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D ((F \<epsilon> \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F f)) \<cdot>\<^sub>D
	\<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D
	((F f \<star>\<^sub>D D.inv (\<Phi> (g, f))) \<cdot>\<^sub>D (F f \<star>\<^sub>D F \<eta>)) \<cdot>\<^sub>D
	D.inv (\<Phi> (f, src\<^sub>C f))"
	proof -
	have "F (\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, f) = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (F \<epsilon> \<star>\<^sub>D F f)"
	using assms \<Phi>.naturality [of "(\<epsilon>, f)"] FF_def C.VV.arr_char
	C.VV.dom_char C.VV.cod_char
	by simp
	moreover have "D.inv (\<Phi> (f, g \<star>\<^sub>C f)) \<cdot>\<^sub>D F (f \<star>\<^sub>C \<eta>) =
	(F f \<star>\<^sub>D F \<eta>) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f))"
	proof -
	have "F (f \<star>\<^sub>C \<eta>) \<cdot>\<^sub>D \<Phi> (f, src\<^sub>C f) = \<Phi> (f, g \<star>\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D F \<eta>)"
	using assms \<Phi>.naturality [of "(f, \<eta>)"] FF_def C.VV.arr_char A.antipar
	C.VV.dom_char C.VV.cod_char
	by simp
	thus ?thesis
	using assms A.antipar cmp_components_are_iso C.VV.arr_char cmp_in_hom
	FF_def C.VV.dom_simp C.VV.cod_simp
	D.invert_opposite_sides_of_square
	[of "\<Phi> (f, g \<star>\<^sub>C f)" "F f \<star>\<^sub>D F \<eta>" "F (f \<star>\<^sub>C \<eta>)" "\<Phi> (f, src\<^sub>C f)"]
	by fastforce
	qed
	ultimately show ?thesis
	using D.comp_assoc by simp
	qed
	also have "... = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D
	\<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D
	(F f \<star>\<^sub>D D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta>) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f))"
	using assms A.antipar cmp_in_hom A.ide_left A.ide_right A'.ide_left A'.ide_right
	D.whisker_left [of "F f" "D.inv (\<Phi> (g, f))" "F \<eta>"]
	D.whisker_right [of "F f" "F \<epsilon>" "\<Phi> (f, g)"]
	by (metis A'.counit_in_vhom A'.unit_simps(1)D.arrI D.comp_assoc
	D.src.preserves_reflects_arr D.src_vcomp D.vseq_implies_hpar(1) cmp_simps(2))
	also have "... = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<cdot>\<^sub>D ?\<epsilon>' \<star>\<^sub>D F f) \<cdot>\<^sub>D
	\<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D
	(F f \<star>\<^sub>D ?\<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f))"
	proof -
	have "F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) = unit (trg\<^sub>C f) \<cdot>\<^sub>D ?\<epsilon>'"
	proof -
	have "D.iso (unit (trg\<^sub>C f))"
	using A.ide_left C.ideD(1) unit_char(2) by blast
	thus ?thesis
	by (metis A'.counit_simps(1) D.comp_assoc D.comp_cod_arr D.inv_is_inverse
	D.seqE D.comp_arr_inv)
	qed
	moreover have "D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> = ?\<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f))"
	using assms(2) unit_char D.comp_arr_inv D.inv_is_inverse D.comp_assoc D.comp_cod_arr
	by (metis A'.unit_simps(1) A.antipar(1) C.ideD(1) C.obj_trg
	D.invert_side_of_triangle(2))
	ultimately show ?thesis by simp
	qed
	also have "... = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D ((unit (trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D
	(?\<epsilon>' \<star>\<^sub>D F f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D ((F f \<star>\<^sub>D ?\<eta>') \<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 A.antipar A'.antipar unit_char D.whisker_left D.whisker_right
	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
	((?\<epsilon>' \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D ?\<eta>')) \<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 D.comp_assoc 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]) \<cdot>\<^sub>D
	\<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 A'.triangle_left D.comp_assoc by simp
	also have "... = F \<l>\<^sub>C\<^sup>-\<^sup>1[f] \<cdot>\<^sub>D F \<r>\<^sub>C[f]"
	using assms A.antipar preserves_lunit(2) preserves_runit(1) by simp
	also have "... = F (\<l>\<^sub>C\<^sup>-\<^sup>1[f] \<cdot>\<^sub>C \<r>\<^sub>C[f])"
	using assms by simp
	finally show ?thesis by simp
	qed
	ultimately show ?thesis
	using is_faithful by blast
	qed
	show "(g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>C \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>C (\<eta> \<star>\<^sub>C g) = \<r>\<^sub>C\<^sup>-\<^sup>1[g] \<cdot>\<^sub>C \<l>\<^sub>C[g]"
	proof -
	have 1: "C.par ((g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>C \<a>\<^sub>C g f g \<cdot>\<^sub>C (\<eta> \<star>\<^sub>C g)) (\<r>\<^sub>C\<^sup>-\<^sup>1[g] \<cdot>\<^sub>C \<l>\<^sub>C[g])"
	using assms A.antipar by auto
	moreover have "F ((g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>C \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>C (\<eta> \<star>\<^sub>C g)) = F (\<r>\<^sub>C\<^sup>-\<^sup>1[g] \<cdot>\<^sub>C \<l>\<^sub>C[g])"
	proof -
	have "F ((g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>C \<a>\<^sub>C g f g \<cdot>\<^sub>C (\<eta> \<star>\<^sub>C g)) =
	F (g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>D F \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>D F (\<eta> \<star>\<^sub>C g)"
	using 1 by auto
	also have "... = (F (g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>D \<Phi> (g, f \<star>\<^sub>C g)) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D
	\<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D
	(D.inv (\<Phi> (g, f)) \<star>\<^sub>D F g) \<cdot>\<^sub>D (D.inv (\<Phi> (g \<star>\<^sub>C f, g)) \<cdot>\<^sub>D F (\<eta> \<star>\<^sub>C g))"
	using assms A.antipar preserves_assoc(1) [of g f g] D.comp_assoc by auto
	also have "... = \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D ((F g \<star>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g))) \<cdot>\<^sub>D
	\<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D
	((D.inv (\<Phi> (g, f)) \<star>\<^sub>D F g) \<cdot>\<^sub>D (F \<eta> \<star>\<^sub>D F g)) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C g, g))"
	proof -
	have "F (g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>D \<Phi> (g, f \<star>\<^sub>C g) = \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D F \<epsilon>)"
	using assms \<Phi>.naturality [of "(g, \<epsilon>)"] FF_def C.VV.arr_char
	C.VV.dom_simp C.VV.cod_simp
	by auto
	moreover have "D.inv (\<Phi> (g \<star>\<^sub>C f, g)) \<cdot>\<^sub>D F (\<eta> \<star>\<^sub>C g) =
	(F \<eta> \<star>\<^sub>D F g) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C g, g))"
	proof -
	have "F (\<eta> \<star>\<^sub>C g) \<cdot>\<^sub>D \<Phi> (trg\<^sub>C g, g) = \<Phi> (g \<star>\<^sub>C f, g) \<cdot>\<^sub>D (F \<eta> \<star>\<^sub>D F g)"
	using assms \<Phi>.naturality [of "(\<eta>, g)"] FF_def C.VV.arr_char A.antipar
	C.VV.dom_simp C.VV.cod_simp
	by auto
	thus ?thesis
	using assms A.antipar cmp_components_are_iso C.VV.arr_char FF_def
	C.VV.dom_simp C.VV.cod_simp
	D.invert_opposite_sides_of_square
	[of "\<Phi> (g \<star>\<^sub>C f, g)" "F \<eta> \<star>\<^sub>D F g" "F (\<eta> \<star>\<^sub>C g)" "\<Phi> (trg\<^sub>C g, g)"]
	by fastforce
	qed
	ultimately show ?thesis
	using D.comp_assoc by simp
	qed
	also have " ... = \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D
	\<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D
	(D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<star>\<^sub>D F g) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C g, g))"
	proof -
	have "(F g \<star>\<^sub>D F \<epsilon>) \<cdot>\<^sub>D (F g \<star>\<^sub>D \<Phi> (f, g)) = F g \<star>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)"
	using assms A.antipar D.whisker_left
	by (metis A'.counit_simps(1) A'.ide_right D.seqE)
	moreover have "(D.inv (\<Phi> (g, f)) \<star>\<^sub>D F g) \<cdot>\<^sub>D (F \<eta> \<star>\<^sub>D F g) =
	D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<star>\<^sub>D F g"
	using assms A.antipar D.whisker_right by simp
	ultimately show ?thesis by simp
	qed
	also have "... = \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (trg\<^sub>C f) \<cdot>\<^sub>D ?\<epsilon>') \<cdot>\<^sub>D
	\<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D
	(?\<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f)) \<star>\<^sub>D F g) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C g, g))"
	proof -
	have "F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) = unit (trg\<^sub>C f) \<cdot>\<^sub>D ?\<epsilon>'"
	using unit_char D.comp_arr_inv D.inv_is_inverse D.comp_assoc D.comp_cod_arr
	by (metis A'.counit_simps(1) C.ideD(1) C.obj_trg D.seqE assms(1))
	moreover have "D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> = ?\<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f))"
	using unit_char D.comp_arr_inv D.inv_is_inverse D.comp_assoc D.comp_cod_arr
	by (metis A'.unit_simps(1) A.unit_simps(1) A.unit_simps(5)
	C.obj_trg D.invert_side_of_triangle(2))
	ultimately show ?thesis by simp
	qed
	also have "... = \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (trg\<^sub>C f)) \<cdot>\<^sub>D
	((F g \<star>\<^sub>D ?\<epsilon>') \<cdot>\<^sub>D \<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D (?\<eta>' \<star>\<^sub>D F g)) \<cdot>\<^sub>D
	(D.inv (unit (src\<^sub>C f)) \<star>\<^sub>D F g) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C g, g))"
	using assms A.antipar unit_char D.whisker_left D.whisker_right D.comp_assoc
	by simp
	also have "... = \<Phi> (g, src\<^sub>C g) \<cdot>\<^sub>D (F g \<star>\<^sub>D unit (trg\<^sub>C f)) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F g] \<cdot>\<^sub>D
	\<l>\<^sub>D[F g] \<cdot>\<^sub>D (D.inv (unit (src\<^sub>C f)) \<star>\<^sub>D F g) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C g, g))"
	using A'.triangle_right D.comp_assoc by simp
	also have "... = F \<r>\<^sub>C\<^sup>-\<^sup>1[g] \<cdot>\<^sub>D F \<l>\<^sub>C[g]"
	using assms A.antipar preserves_lunit(1) preserves_runit(2) D.comp_assoc
	by simp
	also have "... = F (\<r>\<^sub>C\<^sup>-\<^sup>1[g] \<cdot>\<^sub>C \<l>\<^sub>C[g])"
	using assms by simp
	finally show ?thesis by simp
	qed
	ultimately show ?thesis
	using is_faithful by blast
	qed
	qed
	qed

	lemma reflects_adjoint_pair:
	assumes "C.ide f" and "C.ide g"
	and "src\<^sub>C f = trg\<^sub>C g" and "src\<^sub>C g = trg\<^sub>C f"
	and "D.adjoint_pair (F f) (F g)"
	shows "C.adjoint_pair f g"
	proof -
	obtain \<eta>' \<epsilon>' where A': "adjunction_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g) \<eta>' \<epsilon>'"
	using assms D.adjoint_pair_def by auto
	interpret A': adjunction_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> \<eta>' \<epsilon>'
	using A' by auto
	have 1: "\<guillemotleft>\<Phi> (g, f) \<cdot>\<^sub>D \<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f)) : F (src\<^sub>C f) \<Rightarrow>\<^sub>D F (g \<star>\<^sub>C f)\<guillemotright>"
	using assms unit_char [of "src\<^sub>C f"] A'.unit_in_hom
	by (intro D.comp_in_homI, auto)
	have 2: "\<guillemotleft>unit (trg\<^sub>C f) \<cdot>\<^sub>D \<epsilon>' \<cdot>\<^sub>D D.inv (\<Phi> (f, g)): F (f \<star>\<^sub>C g) \<Rightarrow>\<^sub>D F (trg\<^sub>C f)\<guillemotright>"
	using assms cmp_in_hom [of f g] unit_char [of "trg\<^sub>C f"] A'.counit_in_hom
	by (intro D.comp_in_homI, auto)
	obtain \<eta> where \<eta>: "\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright> \<and>
	F \<eta> = \<Phi> (g, f) \<cdot>\<^sub>D \<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f))"
	using assms 1 A'.unit_in_hom cmp_in_hom locally_full by fastforce
	have \<eta>': "\<eta>' = D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)"
	using assms 1 \<eta> cmp_in_hom D.iso_inv_iso cmp_components_are_iso unit_char(2)
	D.invert_side_of_triangle(1) [of "F \<eta>" "\<Phi> (g, f)" "\<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f))"]
	D.invert_side_of_triangle(2) [of "D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta>" \<eta>' "D.inv (unit (src\<^sub>C f))"]
	by (metis (no_types, lifting) C.ideD(1) C.obj_trg D.arrI D.comp_assoc D.inv_inv)
	obtain \<epsilon> where \<epsilon>: "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C trg\<^sub>C f\<guillemotright> \<and>
	F \<epsilon> = unit (trg\<^sub>C f) \<cdot>\<^sub>D \<epsilon>' \<cdot>\<^sub>D D.inv (\<Phi> (f, g))"
	using assms 2 A'.counit_in_hom cmp_in_hom locally_full by fastforce
	have \<epsilon>': "\<epsilon>' = D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)"
	using assms 2 \<epsilon> cmp_in_hom D.iso_inv_iso unit_char(2) D.comp_assoc
	D.invert_side_of_triangle(1) [of "F \<epsilon>" "unit (trg\<^sub>C f)" "\<epsilon>' \<cdot>\<^sub>D D.inv (\<Phi> (f, g))"]
	D.invert_side_of_triangle(2) [of "D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon>" \<epsilon>' "D.inv (\<Phi> (f, g))"]
	by (metis (no_types, lifting) C.arrI C.ideD(1) C.obj_trg D.inv_inv cmp_components_are_iso
	preserves_arr)
	have "adjunction_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 A'.adjunction_in_bicategory_axioms \<eta>' \<epsilon>' by simp
	hence "adjunction_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 \<eta> \<epsilon> reflects_adjunction by simp
	thus ?thesis
	using C.adjoint_pair_def by auto
	qed

	lemma reflects_left_adjoint:
	assumes "C.ide f" and "D.is_left_adjoint (F f)"
	shows "C.is_left_adjoint f"
	proof -
	obtain g' where g': "D.adjoint_pair (F f) g'"
	using assms D.adjoint_pair_def 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 g' locally_essentially_surjective [of "trg\<^sub>C f" "src\<^sub>C f" g']
	D.adjoint_pair_antipar [of "F f" g']
	by auto
	obtain \<phi> where \<phi>: "\<guillemotleft>\<phi> : g' \<Rightarrow>\<^sub>D F g\<guillemotright> \<and> D.iso \<phi>"
	using g D.isomorphic_def D.isomorphic_symmetric by metis
	have "D.adjoint_pair (F f) (F g)"
	using assms g g' \<phi> D.adjoint_pair_preserved_by_iso [of "F f" g' "F f" "F f" \<phi> "F g"]
	by auto
	thus ?thesis
	using assms g reflects_adjoint_pair [of f g] D.adjoint_pair_antipar C.in_hhom_def
	by auto
	qed

	lemma reflects_right_adjoint:
	assumes "C.ide g" and "D.is_right_adjoint (F g)"
	shows "C.is_right_adjoint g"
	proof -
	obtain f' where f': "D.adjoint_pair f' (F g)"
	using assms D.adjoint_pair_def by auto
	obtain f where f: "\<guillemotleft>f : trg\<^sub>C g \<rightarrow>\<^sub>C src\<^sub>C g\<guillemotright> \<and> C.ide f \<and> D.isomorphic (F f) f'"
	using assms f' locally_essentially_surjective [of "trg\<^sub>C g" "src\<^sub>C g" f']
	D.adjoint_pair_antipar [of f' "F g"]
	by auto
	obtain \<phi> where \<phi>: "\<guillemotleft>\<phi> : f' \<Rightarrow>\<^sub>D F f\<guillemotright> \<and> D.iso \<phi>"
	using f D.isomorphic_def D.isomorphic_symmetric by metis
	have "D.adjoint_pair (F f) (F g)"
	using assms f f' \<phi> D.adjoint_pair_preserved_by_iso [of f' "F g" \<phi> "F f" "F g" "F g"]
	by auto
	thus ?thesis
	using assms f reflects_adjoint_pair [of f g] D.adjoint_pair_antipar C.in_hhom_def
	by auto
	qed

	end

	subsection "Composition of Adjunctions"

	text \<open>
	We first consider the strict case, then extend to all bicategories using strictification.
	\<close>

	locale composite_adjunction_in_strict_bicategory =
	strict_bicategory V H \<a> \<i> src trg +
	fg: adjunction_in_strict_bicategory V H \<a> \<i> src trg f g \<zeta> \<xi> +
	hk: adjunction_in_strict_bicategory V H \<a> \<i> src trg h k \<sigma> \<tau>
	for V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<cdot>" 55)
	and H :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<star>" 53)
	and \<a> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("\<a>[_, _, _]")
	and \<i> :: "'a \<Rightarrow> 'a" ("\<i>[_]")
	and src :: "'a \<Rightarrow> 'a"
	and trg :: "'a \<Rightarrow> 'a"
	and f :: "'a"
	and g :: "'a"
	and \<zeta> :: "'a"
	and \<xi> :: "'a"
	and h :: "'a"
	and k :: "'a"
	and \<sigma> :: "'a"
	and \<tau> :: "'a" +
	assumes composable: "src h = trg f"
	begin

	abbreviation \<eta>
	where "\<eta> \<equiv> (g \<star> \<sigma> \<star> f) \<cdot> \<zeta>"

	abbreviation \<epsilon>
	where "\<epsilon> \<equiv> \<tau> \<cdot> (h \<star> \<xi> \<star> k)"

	interpretation adjunction_data_in_bicategory V H \<a> \<i> src trg \<open>h \<star> f\<close> \<open>g \<star> k\<close> \<eta> \<epsilon>
	proof
	show "ide (h \<star> f)"
	using composable by simp
	show "ide (g \<star> k)"
	using fg.antipar hk.antipar composable by simp
	show "\<guillemotleft>\<eta> : src (h \<star> f) \<Rightarrow> (g \<star> k) \<star> h \<star> f\<guillemotright>"
	proof
	show "\<guillemotleft>\<zeta> : src (h \<star> f) \<Rightarrow> g \<star> f\<guillemotright>"
	using fg.antipar hk.antipar composable \<open>ide (h \<star> f)\<close> by auto
	show "\<guillemotleft>g \<star> \<sigma> \<star> f : g \<star> f \<Rightarrow> (g \<star> k) \<star> h \<star> f\<guillemotright>"
	proof -
	have "\<guillemotleft>g \<star> \<sigma> \<star> f : g \<star> trg f \<star> f \<Rightarrow> g \<star> (k \<star> h) \<star> f\<guillemotright>"
	using fg.antipar hk.antipar composable hk.unit_in_hom
	apply (intro hcomp_in_vhom) by auto
	thus ?thesis
	using hcomp_obj_arr hcomp_assoc by fastforce
	qed
	qed
	show "\<guillemotleft>\<epsilon> : (h \<star> f) \<star> g \<star> k \<Rightarrow> src (g \<star> k)\<guillemotright>"
	proof
	show "\<guillemotleft>h \<star> \<xi> \<star> k : (h \<star> f) \<star> g \<star> k \<Rightarrow> h \<star> k\<guillemotright>"
	proof -
	have "\<guillemotleft>h \<star> \<xi> \<star> k : h \<star> (f \<star> g) \<star> k \<Rightarrow> h \<star> trg f \<star> k\<guillemotright>"
	using composable fg.antipar(1-2) hk.antipar(1) by fastforce
	thus ?thesis
	using fg.antipar hk.antipar composable hk.unit_in_hom hcomp_obj_arr hcomp_assoc
	by simp
	qed
	show "\<guillemotleft>\<tau> : h \<star> k \<Rightarrow> src (g \<star> k)\<guillemotright>"
	using fg.antipar hk.antipar composable hk.unit_in_hom by auto
	qed
	qed

	sublocale adjunction_in_strict_bicategory V H \<a> \<i> src trg \<open>h \<star> f\<close> \<open>g \<star> k\<close> \<eta> \<epsilon>
	proof
	show "(\<epsilon> \<star> h \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> f, g \<star> k, h \<star> f] \<cdot> ((h \<star> f) \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[h \<star> f] \<cdot> \<r>[h \<star> f]"
	proof -
	have "(\<epsilon> \<star> h \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> f, g \<star> k, h \<star> f] \<cdot> ((h \<star> f) \<star> \<eta>) =
	(\<tau> \<cdot> (h \<star> \<xi> \<star> k) \<star> h \<star> f) \<cdot> ((h \<star> f) \<star> (g \<star> \<sigma> \<star> f) \<cdot> \<zeta>)"
	using fg.antipar hk.antipar composable strict_assoc comp_ide_arr
	ide_left ide_right antipar(1) antipar(2)
	by (metis arrI seqE strict_assoc' triangle_in_hom(1))
	also have "... = (\<tau> \<star> h \<star> f) \<cdot> ((h \<star> \<xi> \<star> (k \<star> h) \<star> f) \<cdot> (h \<star> (f \<star> g) \<star> \<sigma> \<star> f)) \<cdot> (h \<star> f \<star> \<zeta>)"
	using fg.antipar hk.antipar composable whisker_left [of "h \<star> f"] whisker_right
	comp_assoc hcomp_assoc
	by simp
	also have "... = (\<tau> \<star> h \<star> f) \<cdot> (h \<star> (\<xi> \<star> (k \<star> h)) \<cdot> ((f \<star> g) \<star> \<sigma>) \<star> f) \<cdot> (h \<star> f \<star> \<zeta>)"
	using fg.antipar hk.antipar composable whisker_left whisker_right hcomp_assoc
	by simp
	also have "... = (\<tau> \<star> h \<star> f) \<cdot> (h \<star> (trg f \<star> \<sigma>) \<cdot> (\<xi> \<star> trg f) \<star> f) \<cdot> (h \<star> f \<star> \<zeta>)"
	using fg.antipar hk.antipar composable comp_arr_dom comp_cod_arr
	interchange [of \<xi> "f \<star> g" "k \<star> h" \<sigma>] interchange [of "trg f" \<xi> \<sigma> "trg f"]
	by (metis fg.counit_simps(1) fg.counit_simps(2) fg.counit_simps(3)
	hk.unit_simps(1) hk.unit_simps(2) hk.unit_simps(3))
	also have "... = (\<tau> \<star> h \<star> f) \<cdot> (h \<star> \<sigma> \<cdot> \<xi> \<star> f) \<cdot> (h \<star> f \<star> \<zeta>)"
	using fg.antipar hk.antipar composable hcomp_obj_arr hcomp_arr_obj
	by (metis fg.counit_simps(1) fg.counit_simps(4) hk.unit_simps(1) hk.unit_simps(5)
	obj_src)
	also have "... = ((\<tau> \<star> h \<star> f) \<cdot> (h \<star> \<sigma> \<star> f)) \<cdot> ((h \<star> \<xi> \<star> f) \<cdot> (h \<star> f \<star> \<zeta>))"
	using fg.antipar hk.antipar composable whisker_left whisker_right comp_assoc
	by simp
	also have "... = ((\<tau> \<star> h) \<cdot> (h \<star> \<sigma>) \<star> f) \<cdot> (h \<star> (\<xi> \<star> f) \<cdot> (f \<star> \<zeta>))"
	using fg.antipar hk.antipar composable whisker_left whisker_right hcomp_assoc
	by simp
	also have "... = h \<star> f"
	using fg.antipar hk.antipar composable fg.triangle_left hk.triangle_left
	by simp
	also have "... = \<l>\<^sup>-\<^sup>1[h \<star> f] \<cdot> \<r>[h \<star> f]"
	using fg.antipar hk.antipar composable strict_lunit' strict_runit by simp
	finally show ?thesis by simp
	qed
	show "((g \<star> k) \<star> \<epsilon>) \<cdot> \<a>[g \<star> k, h \<star> f, g \<star> k] \<cdot> (\<eta> \<star> g \<star> k) = \<r>\<^sup>-\<^sup>1[g \<star> k] \<cdot> \<l>[g \<star> k]"
	proof -
	have "((g \<star> k) \<star> \<epsilon>) \<cdot> \<a>[g \<star> k, h \<star> f, g \<star> k] \<cdot> (\<eta> \<star> g \<star> k) =
	((g \<star> k) \<star> \<tau> \<cdot> (h \<star> \<xi> \<star> k)) \<cdot> ((g \<star> \<sigma> \<star> f) \<cdot> \<zeta> \<star> g \<star> k)"
	using fg.antipar hk.antipar composable strict_assoc comp_ide_arr
	ide_left ide_right
	by (metis antipar(1) antipar(2) arrI seqE triangle_in_hom(2))
	also have "... = (g \<star> k \<star> \<tau>) \<cdot> ((g \<star> (k \<star> h) \<star> \<xi> \<star> k) \<cdot> (g \<star> \<sigma> \<star> (f \<star> g) \<star> k)) \<cdot> (\<zeta> \<star> g \<star> k)"
	using fg.antipar hk.antipar composable whisker_left [of "g \<star> k"] whisker_right
	comp_assoc hcomp_assoc
	by simp
	also have "... = (g \<star> k \<star> \<tau>) \<cdot> (g \<star> ((k \<star> h) \<star> \<xi>) \<cdot> (\<sigma> \<star> f \<star> g) \<star> k) \<cdot> (\<zeta> \<star> g \<star> k)"
	using fg.antipar hk.antipar composable whisker_left whisker_right hcomp_assoc
	by simp
	also have "... = (g \<star> k \<star> \<tau>) \<cdot> (g \<star> (\<sigma> \<star> src g) \<cdot> (src g \<star> \<xi>) \<star> k) \<cdot> (\<zeta> \<star> g \<star> k)"
	using fg.antipar hk.antipar composable interchange [of "k \<star> h" \<sigma> \<xi> "f \<star> g"]
	interchange [of \<sigma> "src g" "src g" \<xi>] comp_arr_dom comp_cod_arr
	by (metis fg.counit_simps(1) fg.counit_simps(2) fg.counit_simps(3)
	hk.unit_simps(1) hk.unit_simps(2) hk.unit_simps(3))
	also have "... = (g \<star> k \<star> \<tau>) \<cdot> (g \<star> \<sigma> \<cdot> \<xi> \<star> k) \<cdot> (\<zeta> \<star> g \<star> k)"
	using fg.antipar hk.antipar composable hcomp_obj_arr [of "src g" \<xi>]
	hcomp_arr_obj [of \<sigma> "src g"]
	by simp
	also have "... = ((g \<star> k \<star> \<tau>) \<cdot> (g \<star> \<sigma> \<star> k)) \<cdot> (g \<star> \<xi> \<star> k) \<cdot> (\<zeta> \<star> g \<star> k)"
	using fg.antipar hk.antipar composable whisker_left whisker_right comp_assoc
	by simp
	also have "... = (g \<star> (k \<star> \<tau>) \<cdot> (\<sigma> \<star> k)) \<cdot> ((g \<star> \<xi>) \<cdot> (\<zeta> \<star> g) \<star> k)"
	using fg.antipar hk.antipar composable whisker_left whisker_right hcomp_assoc
	by simp
	also have "... = g \<star> k"
	using fg.antipar hk.antipar composable fg.triangle_right hk.triangle_right
	by simp
	also have "... = \<r>\<^sup>-\<^sup>1[g \<star> k] \<cdot> \<l>[g \<star> k]"
	using fg.antipar hk.antipar composable strict_lunit strict_runit' by simp
	finally show ?thesis by simp
	qed
	qed

	lemma is_adjunction_in_strict_bicategory:
	shows "adjunction_in_strict_bicategory V H \<a> \<i> src trg (h \<star> f) (g \<star> k) \<eta> \<epsilon>"
	..

	end

	context strict_bicategory
	begin

	lemma left_adjoints_compose:
	assumes "is_left_adjoint f" and "is_left_adjoint f'" and "src f' = trg f"
	shows "is_left_adjoint (f' \<star> f)"
	proof -
	obtain g \<eta> \<epsilon> where fg: "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms adjoint_pair_def by auto
	interpret fg: adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using fg by auto
	obtain g' \<eta>' \<epsilon>' where f'g': "adjunction_in_bicategory V H \<a> \<i> src trg f' g' \<eta>' \<epsilon>'"
	using assms adjoint_pair_def by auto
	interpret f'g': adjunction_in_bicategory V H \<a> \<i> src trg f' g' \<eta>' \<epsilon>'
	using f'g' by auto
	interpret f'fgg': composite_adjunction_in_strict_bicategory V H \<a> \<i> src trg
	f g \<eta> \<epsilon> f' g' \<eta>' \<epsilon>'
	using assms apply unfold_locales by simp
	have "adjoint_pair (f' \<star> f) (g \<star> g')"
	using adjoint_pair_def f'fgg'.adjunction_in_bicategory_axioms by auto
	thus ?thesis by auto
	qed

	lemma right_adjoints_compose:
	assumes "is_right_adjoint g" and "is_right_adjoint g'" and "src g = trg g'"
	shows "is_right_adjoint (g \<star> g')"
	proof -
	obtain f \<eta> \<epsilon> where fg: "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms adjoint_pair_def by auto
	interpret fg: adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using fg by auto
	obtain f' \<eta>' \<epsilon>' where f'g': "adjunction_in_bicategory V H \<a> \<i> src trg f' g' \<eta>' \<epsilon>'"
	using assms adjoint_pair_def by auto
	interpret f'g': adjunction_in_bicategory V H \<a> \<i> src trg f' g' \<eta>' \<epsilon>'
	using f'g' by auto
	interpret f'fgg': composite_adjunction_in_strict_bicategory V H \<a> \<i> src trg
	f g \<eta> \<epsilon> f' g' \<eta>' \<epsilon>'
	using assms fg.antipar f'g'.antipar apply unfold_locales by simp
	have "adjoint_pair (f' \<star> f) (g \<star> g')"
	using adjoint_pair_def f'fgg'.adjunction_in_bicategory_axioms by auto
	thus ?thesis by auto
	qed

	end

	text \<open>
	We now use strictification to extend the preceding results to an arbitrary bicategory.
	We only prove that ``left adjoints compose'' and ``right adjoints compose'';
	I did not work out formulas for the unit and counit of the composite adjunction in the
	non-strict case.
	\<close>

	context bicategory
	begin

	interpretation S: strictified_bicategory V H \<a> \<i> src trg ..

	notation S.vcomp (infixr "\<cdot>\<^sub>S" 55)
	notation S.hcomp (infixr "\<star>\<^sub>S" 53)
	notation S.in_hom ("\<guillemotleft>_ : _ \<Rightarrow>\<^sub>S _\<guillemotright>")
	notation S.in_hhom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>S _\<guillemotright>")

	interpretation UP: fully_faithful_functor V S.vcomp S.UP
	using S.UP_is_fully_faithful_functor by auto
	interpretation UP: equivalence_pseudofunctor V H \<a> \<i> src trg
	S.vcomp S.hcomp S.\<a> S.\<i> S.src S.trg S.UP S.cmp\<^sub>U\<^sub>P
	using S.UP_is_equivalence_pseudofunctor by auto

	lemma left_adjoints_compose:
	assumes "is_left_adjoint f" and "is_left_adjoint f'" and "src f = trg f'"
	shows "is_left_adjoint (f \<star> f')"
	proof -
	have "S.is_left_adjoint (S.UP f) \<and> S.is_left_adjoint (S.UP f')"
	using assms UP.preserves_left_adjoint by simp
	moreover have "S.src (S.UP f) = S.trg (S.UP f')"
	using assms left_adjoint_is_ide by simp
	ultimately have "S.is_left_adjoint (S.hcomp (S.UP f) (S.UP f'))"
	using S.left_adjoints_compose by simp
	moreover have "S.isomorphic (S.hcomp (S.UP f) (S.UP f')) (S.UP (f \<star> f'))"
	proof -
	have "\<guillemotleft>S.cmp\<^sub>U\<^sub>P (f, f') : S.hcomp (S.UP f) (S.UP f') \<Rightarrow>\<^sub>S S.UP (f \<star> f')\<guillemotright>"
	using assms left_adjoint_is_ide UP.cmp_in_hom by simp
	moreover have "S.iso (S.cmp\<^sub>U\<^sub>P (f, f'))"
	using assms left_adjoint_is_ide by simp
	ultimately show ?thesis
	using S.isomorphic_def by blast
	qed
	ultimately have "S.is_left_adjoint (S.UP (f \<star> f'))"
	using S.left_adjoint_preserved_by_iso S.isomorphic_def by blast
	thus "is_left_adjoint (f \<star> f')"
	using assms left_adjoint_is_ide UP.reflects_left_adjoint by simp
	qed

	lemma right_adjoints_compose:
	assumes "is_right_adjoint g" and "is_right_adjoint g'" and "src g' = trg g"
	shows "is_right_adjoint (g' \<star> g)"
	proof -
	have "S.is_right_adjoint (S.UP g) \<and> S.is_right_adjoint (S.UP g')"
	using assms UP.preserves_right_adjoint by simp
	moreover have "S.src (S.UP g') = S.trg (S.UP g)"
	using assms right_adjoint_is_ide by simp
	ultimately have "S.is_right_adjoint (S.hcomp (S.UP g') (S.UP g))"
	using S.right_adjoints_compose by simp
	moreover have "S.isomorphic (S.hcomp (S.UP g') (S.UP g)) (S.UP (g' \<star> g))"
	proof -
	have "\<guillemotleft>S.cmp\<^sub>U\<^sub>P (g', g) : S.hcomp (S.UP g') (S.UP g) \<Rightarrow>\<^sub>S S.UP (g' \<star> g)\<guillemotright>"
	using assms right_adjoint_is_ide UP.cmp_in_hom by simp
	moreover have "S.iso (S.cmp\<^sub>U\<^sub>P (g', g))"
	using assms right_adjoint_is_ide by simp
	ultimately show ?thesis
	using S.isomorphic_def by blast
	qed
	ultimately have "S.is_right_adjoint (S.UP (g' \<star> g))"
	using S.right_adjoint_preserved_by_iso S.isomorphic_def by blast
	thus "is_right_adjoint (g' \<star> g)"
	using assms right_adjoint_is_ide UP.reflects_right_adjoint by simp
	qed

	end

	subsection "Choosing Right Adjoints"

	text \<open>
	It will be useful in various situations to suppose that we have made a choice of
	right adjoint for each left adjoint ({\it i.e.} each ``map'') in a bicategory.
	\<close>

	locale chosen_right_adjoints =
	bicategory
	begin

	(* Global notation is evil! *)
	no_notation Transitive_Closure.rtrancl ("(_\<^sup>*)" [1000] 999)

	definition some_right_adjoint ("_\<^sup>*" [1000] 1000)
	where "f\<^sup>* \<equiv> SOME g. adjoint_pair f g"

	definition some_unit
	where "some_unit f \<equiv> SOME \<eta>. \<exists>\<epsilon>. adjunction_in_bicategory V H \<a> \<i> src trg f f\<^sup>* \<eta> \<epsilon>"

	definition some_counit
	where "some_counit f \<equiv>
	SOME \<epsilon>. adjunction_in_bicategory V H \<a> \<i> src trg f f\<^sup>* (some_unit f) \<epsilon>"

	lemma left_adjoint_extends_to_adjunction:
	assumes "is_left_adjoint f"
	shows "adjunction_in_bicategory V H \<a> \<i> src trg f f\<^sup>* (some_unit f) (some_counit f)"
	using assms some_right_adjoint_def adjoint_pair_def some_unit_def some_counit_def
	someI_ex [of "\<lambda>g. adjoint_pair f g"]
	someI_ex [of "\<lambda>\<eta>. \<exists>\<epsilon>. adjunction_in_bicategory V H \<a> \<i> src trg f f\<^sup>* \<eta> \<epsilon>"]
	someI_ex [of "\<lambda>\<epsilon>. adjunction_in_bicategory V H \<a> \<i> src trg f f\<^sup>* (some_unit f) \<epsilon>"]
	by auto

	lemma left_adjoint_extends_to_adjoint_pair:
	assumes "is_left_adjoint f"
	shows "adjoint_pair f f\<^sup>*"
	using assms adjoint_pair_def left_adjoint_extends_to_adjunction by blast

	lemma right_adjoint_in_hom [intro]:
	assumes "is_left_adjoint f"
	shows "\<guillemotleft>f\<^sup>* : trg f \<rightarrow> src f\<guillemotright>"
	and "\<guillemotleft>f\<^sup>* : f\<^sup>* \<Rightarrow> f\<^sup>*\<guillemotright>"
	using assms left_adjoint_extends_to_adjoint_pair adjoint_pair_antipar [of f "f\<^sup>*"]
	by auto

	lemma right_adjoint_simps [simp]:
	assumes "is_left_adjoint f"
	shows "ide f\<^sup>*"
	and "src f\<^sup>* = trg f" and "trg f\<^sup>* = src f"
	and "dom f\<^sup>* = f\<^sup>" and "cod f\<^sup> = f\<^sup>*"
	using assms right_adjoint_in_hom left_adjoint_extends_to_adjoint_pair apply auto
	using assms right_adjoint_is_ide [of "f\<^sup>*"] by blast

	end

	locale map_in_bicategory =
	bicategory + chosen_right_adjoints +
	fixes f :: 'a
	assumes is_map: "is_left_adjoint f"
	begin

	abbreviation \<eta>
	where "\<eta> \<equiv> some_unit f"

	abbreviation \<epsilon>
	where "\<epsilon> \<equiv> some_counit f"

	sublocale adjunction_in_bicategory V H \<a> \<i> src trg f \<open>f\<^sup>*\<close> \<eta> \<epsilon>
	using is_map left_adjoint_extends_to_adjunction by simp

	end

	subsection "Equivalences Refine to Adjoint Equivalences"

	text \<open>
	In this section, we show that, just as an equivalence between categories can always
	be refined to an adjoint equivalence, an internal equivalence in a bicategory can also
	always be so refined.
	The proof, which follows that of Theorem 3.3 from \cite{nlab-adjoint-equivalence},
	makes use of the fact that if an internal equivalence satisfies one of the triangle
	identities, then it also satisfies the other.
	\<close>

	locale adjoint_equivalence_in_bicategory =
	equivalence_in_bicategory +
	adjunction_in_bicategory
	begin

	lemma dual_adjoint_equivalence:
	shows "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg g f (inv \<epsilon>) (inv \<eta>)"
	proof -
	interpret gf: equivalence_in_bicategory V H \<a> \<i> src trg g f \<open>inv \<epsilon>\<close> \<open>inv \<eta>\<close>
	using dual_equivalence by simp
	show ?thesis
	proof
	show "(inv \<eta> \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[g, f, g] \<cdot> (g \<star> inv \<epsilon>) = \<l>\<^sup>-\<^sup>1[g] \<cdot> \<r>[g]"
	proof -
	have "(inv \<eta> \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[g, f, g] \<cdot> (g \<star> inv \<epsilon>) =
	inv ((g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g))"
	using antipar inv_comp isos_compose comp_assoc by simp
	also have "... = inv (\<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g])"
	using triangle_right by simp
	also have "... = \<l>\<^sup>-\<^sup>1[g] \<cdot> \<r>[g]"
	using inv_comp by simp
	finally show ?thesis
	by blast
	qed
	show "(f \<star> inv \<eta>) \<cdot> \<a>[f, g, f] \<cdot> (inv \<epsilon> \<star> f) = \<r>\<^sup>-\<^sup>1[f] \<cdot> \<l>[f]"
	proof -
	have "(f \<star> inv \<eta>) \<cdot> \<a>[f, g, f] \<cdot> (inv \<epsilon> \<star> f) =
	inv ((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>))"
	using antipar inv_comp isos_compose comp_assoc by simp
	also have "... = inv (\<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f])"
	using triangle_left by simp
	also have "... = \<r>\<^sup>-\<^sup>1[f] \<cdot> \<l>[f]"
	using inv_comp by simp
	finally show ?thesis by blast
	qed
	qed
	qed

	end

	context bicategory
	begin

	lemma adjoint_equivalence_preserved_by_iso_right:
	assumes "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	and "\<guillemotleft>\<phi> : g \<Rightarrow> g'\<guillemotright>" and "iso \<phi>"
	shows "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g' ((\<phi> \<star> f) \<cdot> \<eta>) (\<epsilon> \<cdot> (f \<star> inv \<phi>))"
	proof -
	interpret fg: adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using assms by simp
	interpret fg': adjunction_in_bicategory V H \<a> \<i> src trg f g' \<open>(\<phi> \<star> f) \<cdot> \<eta>\<close> \<open>\<epsilon> \<cdot> (f \<star> inv \<phi>)\<close>
	using assms fg.adjunction_in_bicategory_axioms adjunction_preserved_by_iso_right
	by simp
	interpret fg': equivalence_in_bicategory V H \<a> \<i> src trg f g' \<open>(\<phi> \<star> f) \<cdot> \<eta>\<close> \<open>\<epsilon> \<cdot> (f \<star> inv \<phi>)\<close>
	using assms fg.equivalence_in_bicategory_axioms equivalence_preserved_by_iso_right
	by simp
	show ?thesis ..
	qed

	lemma adjoint_equivalence_preserved_by_iso_left:
	assumes "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>" and "iso \<phi>"
	shows "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f' g ((g \<star> \<phi>) \<cdot> \<eta>) (\<epsilon> \<cdot> (inv \<phi> \<star> g))"
	proof -
	interpret fg: adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using assms by simp
	interpret fg': adjunction_in_bicategory V H \<a> \<i> src trg f' g \<open>(g \<star> \<phi>) \<cdot> \<eta>\<close> \<open>\<epsilon> \<cdot> (inv \<phi> \<star> g)\<close>
	using assms fg.adjunction_in_bicategory_axioms adjunction_preserved_by_iso_left
	by simp
	interpret fg': equivalence_in_bicategory V H \<a> \<i> src trg f' g \<open>(g \<star> \<phi>) \<cdot> \<eta>\<close> \<open>\<epsilon> \<cdot> (inv \<phi> \<star> g)\<close>
	using assms fg.equivalence_in_bicategory_axioms equivalence_preserved_by_iso_left
	by simp
	show ?thesis ..
	qed

	end

	context strict_bicategory
	begin

	notation isomorphic (infix "\<cong>" 50)

	lemma equivalence_refines_to_adjoint_equivalence:
	assumes "equivalence_map f" and "\<guillemotleft>g : trg f \<rightarrow> src f\<guillemotright>" and "ide g"
	and "\<guillemotleft>\<eta> : src f \<Rightarrow> g \<star> f\<guillemotright>" and "iso \<eta>"
	shows "\<exists>!\<epsilon>. adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	proof -
	obtain g' \<eta>' \<epsilon>' where E': "equivalence_in_bicategory V H \<a> \<i> src trg f g' \<eta>' \<epsilon>'"
	using assms equivalence_map_def by auto
	interpret E': equivalence_in_bicategory V H \<a> \<i> src trg f g' \<eta>' \<epsilon>'
	using E' by auto
	let ?a = "src f" and ?b = "trg f"
	(* TODO: in_homE cannot be applied automatically to a conjunction. Must break down! *)
	have f_in_hhom: "\<guillemotleft>f : ?a \<rightarrow> ?b\<guillemotright>" and ide_f: "ide f"
	using assms equivalence_map_def by auto
	have g_in_hhom: "\<guillemotleft>g : ?b \<rightarrow> ?a\<guillemotright>" and ide_g: "ide g"
	using assms by auto
	have g'_in_hhom: "\<guillemotleft>g' : ?b \<rightarrow> ?a\<guillemotright>" and ide_g': "ide g'"
	using assms f_in_hhom E'.antipar by auto
	have \<eta>_in_hom: "\<guillemotleft>\<eta> : ?a \<Rightarrow> g \<star> f\<guillemotright>" and iso_\<eta>: "iso \<eta>"
	using assms by auto
	have a: "obj ?a" and b: "obj ?b"
	using f_in_hhom by auto
	have \<eta>_in_hhom: "\<guillemotleft>\<eta> : ?a \<rightarrow> ?a\<guillemotright>"
	using a \<eta>_in_hom
	by (metis arrI in_hhom_def obj_simps(2-3) vconn_implies_hpar(1-2))
	text \<open>
	The following is quoted from \cite{nlab-adjoint-equivalence}:
	\begin{quotation}
	``Since \<open>g \<cong> gfg' \<cong> g'\<close>, the isomorphism \<open>fg' \<cong> 1\<close> also induces an isomorphism \<open>fg \<cong> 1\<close>,
	which we denote \<open>\<xi>\<close>. Now \<open>\<eta>\<close> and \<open>\<xi>\<close> may not satisfy the zigzag identities, but if we
	define \<open>\<epsilon>\<close> by \<open>\<xi> \<cdot> (f \<star> \<eta>\<^sup>-\<^sup>1) \<cdot> (f \<star> g \<star> \<xi>\<^sup>-\<^sup>1) : f \<star> g \<Rightarrow> 1\<close>, then we can verify,
	using string diagram notation as above,
	that \<open>\<epsilon>\<close> satisfies one zigzag identity, and hence (by the previous lemma) also the other.

	Finally, if \<open>\<epsilon>': fg \<Rightarrow> 1\<close> is any other isomorphism satisfying the zigzag identities
	with \<open>\<eta>\<close>, then we have:
	\[
	\<open>\<epsilon>' = \<epsilon>' \<cdot> (\<epsilon> f g) \<cdot> (f \<eta> g) = \<epsilon> \<cdot> (f g \<epsilon>') \<cdot> (f \<eta> g) = \<epsilon>\<close>
	\]
	using the interchange law and two zigzag identities. This shows uniqueness.''
	\end{quotation}
	\<close>
	have 1: "g \<cong> g'"
	proof -
	have "g \<cong> g \<star> ?b"
	using assms hcomp_arr_obj isomorphic_reflexive by auto
	also have "... \<cong> g \<star> f \<star> g'"
	using assms f_in_hhom g_in_hhom g'_in_hhom E'.counit_in_vhom E'.counit_is_iso
	isomorphic_def hcomp_ide_isomorphic isomorphic_symmetric
	by (metis E'.counit_simps(5) in_hhomE trg_trg)
	also have "... \<cong> ?a \<star> g'"
	using assms f_in_hhom g_in_hhom g'_in_hhom ide_g' E'.unit_in_vhom E'.unit_is_iso
	isomorphic_def hcomp_isomorphic_ide isomorphic_symmetric
	by (metis hcomp_assoc hcomp_isomorphic_ide in_hhomE src_src)
	also have "... \<cong> g'"
	using assms
	by (simp add: E'.antipar(1) hcomp_obj_arr isomorphic_reflexive)
	finally show ?thesis by blast
	qed
	have "f \<star> g' \<cong> ?b"
	using E'.counit_is_iso isomorphicI [of \<epsilon>'] by auto
	hence 2: "f \<star> g \<cong> ?b"
	using assms 1 ide_f hcomp_ide_isomorphic [of f g g'] isomorphic_transitive
	isomorphic_symmetric
	by (metis in_hhomE)
	obtain \<xi> where \<xi>: "\<guillemotleft>\<xi> : f \<star> g \<Rightarrow> ?b\<guillemotright> \<and> iso \<xi>"
	using 2 by auto
	have \<xi>_in_hom: "\<guillemotleft>\<xi> : f \<star> g \<Rightarrow> ?b\<guillemotright>" and iso_\<xi>: "iso \<xi>"
	using \<xi> by auto
	have \<xi>_in_hhom: "\<guillemotleft>\<xi> : ?b \<rightarrow> ?b\<guillemotright>"
	using b \<xi>_in_hom
	by (metis \<xi> in_hhom_def iso_is_arr obj_simps(2-3) vconn_implies_hpar(1-4))
	text \<open>
	At the time of this writing, the definition of \<open>\<epsilon>\<close> given on nLab
	\cite{nlab-adjoint-equivalence} had an apparent typo:
	the expression \<open>f \<star> g \<star> \<xi>\<^sup>-\<^sup>1\<close> should read \<open>\<xi>\<^sup>-\<^sup>1 \<star> f \<star> g\<close>, as we have used here.
	\<close>
	let ?\<epsilon> = "\<xi> \<cdot> (f \<star> inv \<eta> \<star> g) \<cdot> (inv \<xi> \<star> f \<star> g)"
	have \<epsilon>_in_hom: "\<guillemotleft>?\<epsilon> : f \<star> g \<Rightarrow> ?b\<guillemotright>"
	proof (intro comp_in_homI)
	show "\<guillemotleft>f \<star> inv \<eta> \<star> g : f \<star> g \<star> f \<star> g \<Rightarrow> f \<star> g\<guillemotright>"
	proof -
	have "\<guillemotleft>f \<star> inv \<eta> \<star> g : f \<star> (g \<star> f) \<star> g \<Rightarrow> f \<star> g\<guillemotright>"
	proof -
	have "\<guillemotleft>f \<star> inv \<eta> \<star> g : f \<star> (g \<star> f) \<star> g \<Rightarrow> f \<star> ?a \<star> g\<guillemotright>"
	using assms \<eta>_in_hom iso_\<eta> by (intro hcomp_in_vhom) auto
	thus ?thesis
	using assms f_in_hhom hcomp_obj_arr by (metis in_hhomE)
	qed
	moreover have "f \<star> (g \<star> f) \<star> g = f \<star> g \<star> f \<star> g"
	using hcomp_assoc by simp
	ultimately show ?thesis by simp
	qed
	show "\<guillemotleft>inv \<xi> \<star> f \<star> g : f \<star> g \<Rightarrow> f \<star> g \<star> f \<star> g\<guillemotright>"
	proof -
	have "\<guillemotleft>inv \<xi> \<star> f \<star> g : ?b \<star> f \<star> g \<Rightarrow> (f \<star> g) \<star> f \<star> g\<guillemotright>"
	using assms \<xi>_in_hom iso_\<xi> by (intro hcomp_in_vhom, auto)
	thus ?thesis
	using hcomp_assoc f_in_hhom g_in_hhom b hcomp_obj_arr [of ?b "f \<star> g"]
	by fastforce
	qed
	show "\<guillemotleft>\<xi> : f \<star> g \<Rightarrow> ?b\<guillemotright>"
	using \<xi>_in_hom by blast
	qed
	have "iso ?\<epsilon>"
	using f_in_hhom g_in_hhom \<eta>_in_hhom ide_f ide_g \<eta>_in_hom iso_\<eta> \<xi>_in_hhom \<xi>_in_hom iso_\<xi>
	iso_inv_iso isos_compose
	by (metis \<epsilon>_in_hom arrI hseqE ide_is_iso iso_hcomp seqE)
	have 4: "\<guillemotleft>inv \<xi> \<star> f : ?b \<star> f \<Rightarrow> f \<star> g \<star> f\<guillemotright>"
	proof -
	have "\<guillemotleft>inv \<xi> \<star> f : ?b \<star> f \<Rightarrow> (f \<star> g) \<star> f\<guillemotright>"
	using \<xi>_in_hom iso_\<xi> f_in_hhom
	by (intro hcomp_in_vhom, auto)
	thus ?thesis
	using hcomp_assoc by simp
	qed
	text \<open>
	First show \<open>?\<epsilon>\<close> and \<open>\<eta>\<close> satisfy the ``left'' triangle identity.
	\<close>
	have triangle_left: "(?\<epsilon> \<star> f) \<cdot> (f \<star> \<eta>) = f"
	proof -
	have "(?\<epsilon> \<star> f) \<cdot> (f \<star> \<eta>) = (\<xi> \<star> f) \<cdot> (f \<star> inv \<eta> \<star> g \<star> f) \<cdot> (inv \<xi> \<star> f \<star> g \<star> f) \<cdot> (?b \<star> f \<star> \<eta>)"
	proof -
	have "f \<star> \<eta> = ?b \<star> f \<star> \<eta>"
	using b \<eta>_in_hhom hcomp_obj_arr [of ?b "f \<star> \<eta>"] by fastforce
	moreover have "\<xi> \<cdot> (f \<star> inv \<eta> \<star> g) \<cdot> (inv \<xi> \<star> f \<star> g) \<star> f =
	(\<xi> \<star> f) \<cdot> ((f \<star> inv \<eta> \<star> g) \<star> f) \<cdot> ((inv \<xi> \<star> f \<star> g) \<star> f)"
	using ide_f ide_g \<xi>_in_hhom \<xi>_in_hom iso_\<xi> \<eta>_in_hhom \<eta>_in_hom iso_\<eta> whisker_right
	by (metis \<epsilon>_in_hom arrI seqE)
	moreover have "... = (\<xi> \<star> f) \<cdot> (f \<star> inv \<eta> \<star> g \<star> f) \<cdot> (inv \<xi> \<star> f \<star> g \<star> f)"
	using hcomp_assoc by simp
	ultimately show ?thesis
	using comp_assoc by simp
	qed
	also have "... = (\<xi> \<star> f) \<cdot> ((f \<star> inv \<eta> \<star> g \<star> f) \<cdot> (f \<star> g \<star> f \<star> \<eta>)) \<cdot> (inv \<xi> \<star> f)"
	proof -
	have "(inv \<xi> \<star> f \<star> g \<star> f) \<cdot> (?b \<star> f \<star> \<eta>) = ((inv \<xi> \<star> f) \<star> (g \<star> f)) \<cdot> ((?b \<star> f) \<star> \<eta>)"
	using hcomp_assoc by simp
	also have "... = (inv \<xi> \<star> f) \<cdot> (?b \<star> f) \<star> (g \<star> f) \<cdot> \<eta>"
	proof -
	have "seq (inv \<xi> \<star> f) (?b \<star> f)"
	using a b 4 ide_f ide_g \<xi>_in_hhom \<xi>_in_hom iso_\<xi> by blast
	moreover have "seq (g \<star> f) \<eta>"
	using f_in_hhom g_in_hhom ide_g ide_f \<eta>_in_hom by fast
	ultimately show ?thesis
	using interchange [of "inv \<xi> \<star> f" "?b \<star> f" "g \<star> f" \<eta>] by simp
	qed
	also have "... = inv \<xi> \<star> f \<star> \<eta>"
	using 4 comp_arr_dom comp_cod_arr \<eta>_in_hom hcomp_assoc by (metis in_homE)
	also have "... = (f \<star> g) \<cdot> inv \<xi> \<star> (f \<star> \<eta>) \<cdot> (f \<star> ?a)"
	proof -
	have "(f \<star> g) \<cdot> inv \<xi> = inv \<xi>"
	using \<xi>_in_hom iso_\<xi> comp_cod_arr by auto
	moreover have "(f \<star> \<eta>) \<cdot> (f \<star> ?a) = f \<star> \<eta>"
	proof -
	have "\<guillemotleft>f \<star> \<eta> : f \<star> ?a \<Rightarrow> f \<star> g \<star> f\<guillemotright>"
	using \<eta>_in_hom by fastforce
	thus ?thesis
	using comp_arr_dom by blast
	qed
	ultimately show ?thesis by argo
	qed
	also have "... = ((f \<star> g) \<star> (f \<star> \<eta>)) \<cdot> (inv \<xi> \<star> (f \<star> ?a))"
	proof -
	have "seq (f \<star> g) (inv \<xi>)"
	using \<xi>_in_hom iso_\<xi> comp_cod_arr by auto
	moreover have "seq (f \<star> \<eta>) (f \<star> ?a)"
	using f_in_hhom \<eta>_in_hom by force
	ultimately show ?thesis
	using interchange by simp
	qed
	also have "... = (f \<star> g \<star> f \<star> \<eta>) \<cdot> (inv \<xi> \<star> f)"
	using hcomp_arr_obj hcomp_assoc by auto
	finally have "(inv \<xi> \<star> f \<star> g \<star> f) \<cdot> (?b \<star> f \<star> \<eta>) = (f \<star> g \<star> f \<star> \<eta>) \<cdot> (inv \<xi> \<star> f)"
	by simp
	thus ?thesis
	using comp_assoc by simp
	qed
	also have "... = (\<xi> \<star> f) \<cdot> ((f \<star> ?a \<star> \<eta>) \<cdot> (f \<star> inv \<eta> \<star> ?a)) \<cdot> (inv \<xi> \<star> f)"
	proof -
	have "(f \<star> inv \<eta> \<star> g \<star> f) \<cdot> (f \<star> (g \<star> f) \<star> \<eta>) = f \<star> (inv \<eta> \<star> g \<star> f) \<cdot> ((g \<star> f) \<star> \<eta>)"
	proof -
	have "(f \<star> (inv \<eta> \<star> g) \<star> f) \<cdot> (f \<star> (g \<star> f) \<star> \<eta>) = f \<star> ((inv \<eta> \<star> g) \<star> f) \<cdot> ((g \<star> f) \<star> \<eta>)"
	proof -
	have "seq ((inv \<eta> \<star> g) \<star> f) ((g \<star> f) \<star> \<eta>)"
	proof -
	have "seq (inv \<eta> \<star> g \<star> f) ((g \<star> f) \<star> \<eta>)"
	using f_in_hhom ide_f g_in_hhom ide_g \<eta>_in_hhom \<eta>_in_hom iso_\<eta>
	apply (intro seqI hseqI')
	apply auto
	by fastforce+
	thus ?thesis
	using hcomp_assoc by simp
	qed
	thus ?thesis
	using whisker_left by simp
	qed
	thus ?thesis
	using hcomp_assoc by simp
	qed
	also have "... = f \<star> (?a \<star> \<eta>) \<cdot> (inv \<eta> \<star> ?a)"
	proof -
	have "(inv \<eta> \<star> g \<star> f) \<cdot> ((g \<star> f) \<star> \<eta>) = (?a \<star> \<eta>) \<cdot> (inv \<eta> \<star> ?a)"
	proof -
	have "(inv \<eta> \<star> g \<star> f) \<cdot> ((g \<star> f) \<star> \<eta>) = inv \<eta> \<cdot> (g \<star> f) \<star> (g \<star> f) \<cdot> \<eta>"
	using g_in_hhom ide_g \<eta>_in_hom iso_\<eta>
	interchange [of "inv \<eta>" "g \<star> f" "g \<star> f" \<eta>]
	by force
	also have "... = inv \<eta> \<star> \<eta>"
	using \<eta>_in_hom iso_\<eta> comp_arr_dom comp_cod_arr by auto
	also have "... = ?a \<cdot> inv \<eta> \<star> \<eta> \<cdot> ?a"
	using \<eta>_in_hom iso_\<eta> comp_arr_dom comp_cod_arr by auto
	also have "... = (?a \<star> \<eta>) \<cdot> (inv \<eta> \<star> ?a)"
	using a \<eta>_in_hom iso_\<eta> interchange [of ?a "inv \<eta>" \<eta> ?a] by blast
	finally show ?thesis by simp
	qed
	thus ?thesis by argo
	qed
	also have "... = (f \<star> ?a \<star> \<eta>) \<cdot> (f \<star> inv \<eta> \<star> ?a)"
	proof -
	have "seq (?a \<star> \<eta>) (inv \<eta> \<star> ?a)"
	proof (intro seqI')
	show "\<guillemotleft>inv \<eta> \<star> ?a : (g \<star> f) \<star> ?a \<Rightarrow> ?a \<star> ?a\<guillemotright>"
	using a g_in_hhom \<eta>_in_hom iso_\<eta> hseqI ide_f ide_g
	by (intro hcomp_in_vhom) auto
	show "\<guillemotleft>?a \<star> \<eta> : ?a \<star> ?a \<Rightarrow> ?a \<star> g \<star> f\<guillemotright>"
	using a \<eta>_in_hom hseqI by (intro hcomp_in_vhom) auto
	qed
	thus ?thesis
	using whisker_left by simp
	qed
	finally show ?thesis
	using hcomp_assoc by simp
	qed
	also have "... = (\<xi> \<star> f) \<cdot> ((f \<star> \<eta>) \<cdot> (f \<star> inv \<eta>)) \<cdot> (inv \<xi> \<star> f)"
	using a \<eta>_in_hhom iso_\<eta> hcomp_obj_arr [of ?a \<eta>] hcomp_arr_obj [of "inv \<eta>" ?a] by auto
	also have "... = (\<xi> \<star> f) \<cdot> (inv \<xi> \<star> f)"
	proof -
	have "((f \<star> \<eta>) \<cdot> (f \<star> inv \<eta>)) \<cdot> (inv \<xi> \<star> f) = (f \<star> \<eta> \<cdot> inv \<eta>) \<cdot> (inv \<xi> \<star> f)"
	using \<eta>_in_hhom iso_\<eta> whisker_left inv_in_hom by auto
	also have "... = (f \<star> g \<star> f) \<cdot> (inv \<xi> \<star> f)"
	using \<eta>_in_hom iso_\<eta> comp_arr_inv inv_is_inverse by auto
	also have "... = inv \<xi> \<star> f"
	using 4 comp_cod_arr by blast
	ultimately show ?thesis by simp
	qed
	also have "... = f"
	proof -
	have "(\<xi> \<star> f) \<cdot> (inv \<xi> \<star> f) = \<xi> \<cdot> inv \<xi> \<star> f"
	using \<xi>_in_hhom iso_\<xi> whisker_right by auto
	also have "... = ?b \<star> f"
	using \<xi>_in_hom iso_\<xi> comp_arr_inv' by auto
	also have "... = f"
	using hcomp_obj_arr by auto
	finally show ?thesis by blast
	qed
	finally show ?thesis by blast
	qed

	(* TODO: Putting this earlier breaks some steps in the proof. *)
	interpret E: equivalence_in_strict_bicategory V H \<a> \<i> src trg f g \<eta> ?\<epsilon>
	using ide_g \<eta>_in_hom \<epsilon>_in_hom g_in_hhom \<open>iso \<eta>\<close> \<open>iso ?\<epsilon>\<close>
	by (unfold_locales, auto)

	text \<open>
	Apply ``triangle left if and only iff right'' to show the ``right'' triangle identity.
	\<close>
	have triangle_right: "((g \<star> \<xi> \<cdot> (f \<star> inv \<eta> \<star> g) \<cdot> (inv \<xi> \<star> f \<star> g)) \<cdot> (\<eta> \<star> g) = g)"
	using triangle_left E.triangle_left_iff_right by simp

	text \<open>
	Use the two triangle identities to establish an adjoint equivalence and show that
	there is only one choice for the counit.
	\<close>
	show "\<exists>!\<epsilon>. adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	proof -
	have "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> ?\<epsilon>"
	proof
	show "(?\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
	proof -
	have "(?\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = (?\<epsilon> \<star> f) \<cdot> (f \<star> \<eta>)"
	proof -
	have "seq \<a>\<^sup>-\<^sup>1[f, g, f] (f \<star> \<eta>)"
	using E.antipar
	by (intro seqI, auto)
	hence "\<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = f \<star> \<eta>"
	using ide_f ide_g E.antipar triangle_right strict_assoc' comp_ide_arr
	by presburger
	thus ?thesis by simp
	qed
	also have "... = f"
	using triangle_left by simp
	also have "... = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
	using strict_lunit strict_runit by simp
	finally show ?thesis by simp
	qed
	show "(g \<star> ?\<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
	proof -
	have "(g \<star> ?\<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = (g \<star> ?\<epsilon>) \<cdot> (\<eta> \<star> g)"
	proof -
	have "seq \<a>[g, f, g] (\<eta> \<star> g)"
	using E.antipar
	by (intro seqI, auto)
	hence "\<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<eta> \<star> g"
	using ide_f ide_g E.antipar triangle_right strict_assoc comp_ide_arr
	by presburger
	thus ?thesis by simp
	qed
	also have "... = g"
	using triangle_right by simp
	also have "... = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
	using strict_lunit strict_runit by simp
	finally show ?thesis by blast
	qed
	qed
	moreover have "\<And>\<epsilon> \<epsilon>'. \<lbrakk> adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>;
	adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>' \<rbrakk>
	\<Longrightarrow> \<epsilon> = \<epsilon>'"
	using adjunction_unit_determines_counit
	by (meson adjoint_equivalence_in_bicategory.axioms(2))
	ultimately show ?thesis by auto
	qed
	qed

	end

	text \<open>
	We now apply strictification to generalize the preceding result to an arbitrary bicategory.
	\<close>

	context bicategory
	begin

	interpretation S: strictified_bicategory V H \<a> \<i> src trg ..

	notation S.vcomp (infixr "\<cdot>\<^sub>S" 55)
	notation S.hcomp (infixr "\<star>\<^sub>S" 53)
	notation S.in_hom ("\<guillemotleft>_ : _ \<Rightarrow>\<^sub>S _\<guillemotright>")
	notation S.in_hhom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>S _\<guillemotright>")

	interpretation UP: fully_faithful_functor V S.vcomp S.UP
	using S.UP_is_fully_faithful_functor by auto
	interpretation UP: equivalence_pseudofunctor V H \<a> \<i> src trg
	S.vcomp S.hcomp S.\<a> S.\<i> S.src S.trg S.UP S.cmp\<^sub>U\<^sub>P
	using S.UP_is_equivalence_pseudofunctor by auto
	interpretation UP: pseudofunctor_into_strict_bicategory V H \<a> \<i> src trg
	S.vcomp S.hcomp S.\<a> S.\<i> S.src S.trg S.UP S.cmp\<^sub>U\<^sub>P
	..

	lemma equivalence_refines_to_adjoint_equivalence:
	assumes "equivalence_map f" and "\<guillemotleft>g : trg f \<rightarrow> src f\<guillemotright>" and "ide g"
	and "\<guillemotleft>\<eta> : src f \<Rightarrow> g \<star> f\<guillemotright>" and "iso \<eta>"
	shows "\<exists>!\<epsilon>. adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	proof -
	text \<open>
	To unpack the consequences of the assumptions, we need to obtain an
	interpretation of @{locale equivalence_in_bicategory}, even though we don't
	need the associated data other than \<open>f\<close>, \<open>a\<close>, and \<open>b\<close>.
	\<close>
	obtain g' \<phi> \<psi> where E: "equivalence_in_bicategory V H \<a> \<i> src trg f g' \<phi> \<psi>"
	using assms equivalence_map_def by auto
	interpret E: equivalence_in_bicategory V H \<a> \<i> src trg f g' \<phi> \<psi>
	using E by auto
	let ?a = "src f" and ?b = "trg f"
	have ide_f: "ide f" by simp
	have f_in_hhom: "\<guillemotleft>f : ?a \<rightarrow> ?b\<guillemotright>" by simp
	have a: "obj ?a" and b: "obj ?b" by auto
	have 1: "S.equivalence_map (S.UP f)"
	using assms UP.preserves_equivalence_maps by simp
	let ?\<eta>' = "S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit ?a"
	have 2: "\<guillemotleft>S.UP \<eta> : S.UP ?a \<Rightarrow>\<^sub>S S.UP (g \<star> f)\<guillemotright>"
	using assms UP.preserves_hom [of \<eta> "src f" "g \<star> f"] by auto
	have 3: "\<guillemotleft>?\<eta>' : UP.map\<^sub>0 ?a \<Rightarrow>\<^sub>S S.UP g \<star>\<^sub>S S.UP f\<guillemotright> \<and> S.iso ?\<eta>'"
	proof (intro S.comp_in_homI conjI)
	show "\<guillemotleft>S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) : S.UP (g \<star> f) \<Rightarrow>\<^sub>S S.UP g \<star>\<^sub>S S.UP f\<guillemotright>"
	using assms UP.cmp_in_hom(2) by auto
	moreover show "\<guillemotleft>UP.unit ?a : UP.map\<^sub>0 ?a \<Rightarrow>\<^sub>S S.UP ?a\<guillemotright>" by auto
	moreover show "\<guillemotleft>S.UP \<eta> : S.UP ?a \<Rightarrow>\<^sub>S S.UP (g \<star> f)\<guillemotright>"
	using 2 by simp
	ultimately show "S.iso (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit ?a)"
	using assms UP.cmp_components_are_iso UP.unit_char(2)
	by (intro S.isos_compose) auto
	qed
	have ex_un_\<xi>': "\<exists>!\<xi>'. adjoint_equivalence_in_bicategory S.vcomp S.hcomp S.\<a> S.\<i> S.src S.trg
	(S.UP f) (S.UP g) ?\<eta>' \<xi>'"
	proof -
	have "\<guillemotleft>S.UP g : S.trg (S.UP f) \<rightarrow>\<^sub>S S.src (S.UP f)\<guillemotright>"
	using assms(2) by auto
	moreover have "S.ide (S.UP g)"
	by (simp add: assms(3))
	ultimately show ?thesis
	using 1 3 S.equivalence_refines_to_adjoint_equivalence S.UP_map\<^sub>0_obj by simp
	qed
	obtain \<xi>' where \<xi>': "adjoint_equivalence_in_bicategory S.vcomp S.hcomp S.\<a> S.\<i> S.src S.trg
	(S.UP f) (S.UP g) ?\<eta>' \<xi>'"
	using ex_un_\<xi>' by auto
	interpret E': adjoint_equivalence_in_bicategory S.vcomp S.hcomp S.\<a> S.\<i> S.src S.trg
	\<open>S.UP f\<close> \<open>S.UP g\<close> ?\<eta>' \<xi>'
	using \<xi>' by auto
	let ?\<epsilon>' = "UP.unit ?b \<cdot>\<^sub>S \<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g))"
	have \<epsilon>': "\<guillemotleft>?\<epsilon>' : S.UP (f \<star> g) \<Rightarrow>\<^sub>S S.UP ?b\<guillemotright>"
	using assms(2-3) by auto
	have ex_un_\<epsilon>: "\<exists>!\<epsilon>. \<guillemotleft>\<epsilon> : f \<star> g \<Rightarrow> ?b\<guillemotright> \<and> S.UP \<epsilon> = ?\<epsilon>'"
	proof -
	have "\<exists>\<epsilon>. \<guillemotleft>\<epsilon> : f \<star> g \<Rightarrow> ?b\<guillemotright> \<and> S.UP \<epsilon> = ?\<epsilon>'"
	proof -
	have "src (f \<star> g) = src ?b \<and> trg (f \<star> g) = trg ?b"
	using assms(2) f_in_hhom by auto
	moreover have "ide (f \<star> g)"
	using assms(2-3) by auto
	ultimately show ?thesis
	using \<epsilon>' UP.locally_full by auto
	qed
	moreover have
	"\<And>\<mu> \<nu>. \<lbrakk> \<guillemotleft>\<mu> : f \<star> g \<Rightarrow> ?b\<guillemotright>; S.UP \<mu> = ?\<epsilon>'; \<guillemotleft>\<nu> : f \<star> g \<Rightarrow> ?b\<guillemotright>; S.UP \<nu> = ?\<epsilon>' \<rbrakk>
	\<Longrightarrow> \<mu> = \<nu>"
	proof -
	fix \<mu> \<nu>
	assume \<mu>: "\<guillemotleft>\<mu> : f \<star> g \<Rightarrow> ?b\<guillemotright>" and \<nu>: "\<guillemotleft>\<nu> : f \<star> g \<Rightarrow> ?b\<guillemotright>"
	and 1: "S.UP \<mu> = ?\<epsilon>'" and 2: "S.UP \<nu> = ?\<epsilon>'"
	have "par \<mu> \<nu>"
	using \<mu> \<nu> by fastforce
	thus "\<mu> = \<nu>"
	using 1 2 UP.is_faithful [of \<mu> \<nu>] by simp
	qed
	ultimately show ?thesis by auto
	qed
	have iso_\<epsilon>': "S.iso ?\<epsilon>'"
	proof (intro S.isos_compose)
	show "S.iso (S.inv (S.cmp\<^sub>U\<^sub>P (f, g)))"
	using assms UP.cmp_components_are_iso by auto
	show "S.iso \<xi>'"
	using E'.counit_is_iso by blast
	show "S.iso (UP.unit ?b)"
	using b UP.unit_char(2) by simp
	show "S.seq (UP.unit ?b) (\<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g)))"
	proof (intro S.seqI')
	show "\<guillemotleft>UP.unit ?b : UP.map\<^sub>0 ?b \<Rightarrow>\<^sub>S S.UP ?b\<guillemotright>"
	using b UP.unit_char by simp
	show "\<guillemotleft>\<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g)) : S.UP (f \<star> g) \<Rightarrow>\<^sub>S UP.map\<^sub>0 ?b\<guillemotright>"
	using assms by auto
	qed
	thus "S.seq \<xi>' (S.inv (S.cmp\<^sub>U\<^sub>P (f, g)))" by auto
	qed
	obtain \<epsilon> where \<epsilon>: "\<guillemotleft>\<epsilon> : f \<star> g \<Rightarrow> ?b\<guillemotright> \<and> S.UP \<epsilon> = ?\<epsilon>'"
	using ex_un_\<epsilon> by auto
	interpret E'': equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using assms(1,3-5)
	apply unfold_locales
	apply simp_all
	using assms(2) \<epsilon>
	apply auto[1]
	using \<epsilon> iso_\<epsilon>' UP.reflects_iso [of \<epsilon>]
	by auto
	interpret E'': adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	proof
	show "(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
	proof -
	have "S.UP ((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>)) =
	S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))"
	using E''.UP_triangle(3) by simp
	also have "... = S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S
	(UP.unit ?b \<cdot>\<^sub>S \<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g)) \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))"
	using \<epsilon> S.comp_assoc by simp
	also have "... = S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (UP.unit ?b \<cdot>\<^sub>S \<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) \<cdot>\<^sub>S
	S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))"
	proof -
	have "\<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g)) \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) = \<xi>'"
	proof -
	have "S.iso (S.cmp\<^sub>U\<^sub>P (f, g))"
	using assms by auto
	moreover have "S.dom (S.cmp\<^sub>U\<^sub>P (f, g)) = S.UP f \<star>\<^sub>S S.UP g"
	using assms by auto
	ultimately have "S.inv (S.cmp\<^sub>U\<^sub>P (f, g)) \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) = S.UP f \<star>\<^sub>S S.UP g"
	using S.comp_inv_arr' by simp
	thus ?thesis
	using S.comp_arr_dom E'.counit_in_hom(2) by simp
	qed
	thus ?thesis by argo
	qed
	also have
	"... = S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (UP.unit ?b \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S
	((\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f))) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.UP \<eta>)) \<cdot>\<^sub>S
	S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))"
	proof -
	have "UP.unit ?b \<cdot>\<^sub>S \<xi>' \<star>\<^sub>S S.UP f = (UP.unit ?b \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S (\<xi>' \<star>\<^sub>S S.UP f)"
	using assms b UP.unit_char S.whisker_right S.UP_map\<^sub>0_obj by auto
	moreover have "S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> =
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f))) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.UP \<eta>)"
	using assms S.whisker_left by auto
	ultimately show ?thesis
	using S.comp_assoc by simp
	qed
	also have "... = (S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (UP.unit ?b \<star>\<^sub>S S.UP f)) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (UP.unit (src f))) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))"
	proof -
	have "(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f))) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.UP \<eta>)
	= (S.UP f \<star>\<^sub>S S.inv (UP.unit (src f)))"
	proof -
	have "(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f))) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.UP \<eta>) =
	S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>"
	using assms S.whisker_left by auto
	hence "((\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f))) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.UP \<eta>))
	= ((\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>))"
	by simp
	also have "... = ((\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f)) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>)"
	proof -
	have "(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) = \<xi>' \<star>\<^sub>S S.UP f"
	proof -
	have "\<guillemotleft>\<xi>' \<star>\<^sub>S S.UP f :
	(S.UP f \<star>\<^sub>S S.UP g) \<star>\<^sub>S S.UP f \<Rightarrow>\<^sub>S S.trg (S.UP f) \<star>\<^sub>S S.UP f\<guillemotright>"
	using assms by auto
	moreover have "\<guillemotleft>S.\<a>' (S.UP f) (S.UP g) (S.UP f) :
	S.UP f \<star>\<^sub>S S.UP g \<star>\<^sub>S S.UP f \<Rightarrow>\<^sub>S (S.UP f \<star>\<^sub>S S.UP g) \<star>\<^sub>S S.UP f\<guillemotright>"
	using assms S.assoc'_in_hom by auto
	ultimately show ?thesis
	using assms S.strict_assoc' S.iso_assoc S.hcomp_assoc E'.antipar
	S.comp_arr_ide S.seqI'
	by (metis (no_types, lifting) E'.ide_left E'.ide_right)
	qed
	thus ?thesis
	using S.comp_assoc by simp
	qed
	also have "... = ((\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>))"
	using S.comp_assoc by simp
	also have "... = (S.UP f \<star>\<^sub>S S.inv (UP.unit (src f)))"
	proof -
	have "(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) =
	(S.UP f \<star>\<^sub>S S.inv (UP.unit (src f)))"
	proof -
	have "(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S UP.unit ?a) =
	S.lunit' (S.UP f) \<cdot>\<^sub>S S.runit (S.UP f)"
	proof -
	have "(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S UP.unit ?a) =
	(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit ?a)"
	proof -
	have "S.seq (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) (UP.unit ?a)"
	using assms UP.unit_char UP.cmp_components_are_iso
	E'.unit_simps(1) S.comp_assoc
	by presburger
	hence "(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S UP.unit ?a) =
	S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit ?a"
	using assms UP.unit_char UP.cmp_components_are_iso S.comp_assoc
	S.whisker_left [of "S.UP f" "S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>" "UP.unit ?a"]
	by simp
	thus ?thesis by simp
	qed
	thus ?thesis
	using assms E'.triangle_left UP.cmp_components_are_iso UP.unit_char
	by argo
	qed
	also have "... = S.UP f"
	using S.strict_lunit' S.strict_runit by simp
	finally have 1: "((\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>)) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S UP.unit ?a) = S.UP f"
	using S.comp_assoc by simp
	have "(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) =
	S.UP f \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (UP.unit ?a))"
	proof -
	have "S.arr (S.UP f)"
	using assms by simp
	moreover have "S.iso (S.UP f \<star>\<^sub>S UP.unit ?a)"
	using assms UP.unit_char S.UP_map\<^sub>0_obj by auto
	moreover have "S.inv (S.UP f \<star>\<^sub>S UP.unit ?a) =
	S.UP f \<star>\<^sub>S S.inv (UP.unit ?a)"
	using assms a UP.unit_char S.UP_map\<^sub>0_obj by auto
	ultimately show ?thesis
	using assms 1 UP.unit_char UP.cmp_components_are_iso
	S.invert_side_of_triangle(2)
	[of "S.UP f" "(\<xi>' \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S
	(S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>)"
	"S.UP f \<star>\<^sub>S UP.unit ?a"]
	by presburger
	qed
	also have "... = S.UP f \<star>\<^sub>S S.inv (UP.unit ?a)"
	proof -
	have "\<guillemotleft>S.UP f \<star>\<^sub>S S.inv (UP.unit ?a) :
	S.UP f \<star>\<^sub>S S.UP ?a \<Rightarrow>\<^sub>S S.UP f \<star>\<^sub>S UP.map\<^sub>0 ?a\<guillemotright>"
	using assms ide_f f_in_hhom UP.unit_char [of ?a] S.inv_in_hom
	apply (intro S.hcomp_in_vhom)
	apply auto[1]
	apply blast
	by auto
	moreover have "S.UP f \<star>\<^sub>S UP.map\<^sub>0 ?a = S.UP f"
	using a S.hcomp_arr_obj S.UP_map\<^sub>0_obj by auto
	finally show ?thesis
	using S.comp_cod_arr by blast
	qed
	finally show ?thesis by auto
	qed
	thus ?thesis
	using S.comp_assoc by simp
	qed
	finally show ?thesis by simp
	qed
	thus ?thesis
	using S.comp_assoc by simp
	qed
	also have "... = S.UP \<l>\<^sup>-\<^sup>1[f] \<cdot>\<^sub>S S.UP \<r>[f]"
	proof -
	have "S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (UP.unit ?b \<star>\<^sub>S S.UP f) = S.UP \<l>\<^sup>-\<^sup>1[f]"
	proof -
	have "S.UP f = S.UP \<l>[f] \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (UP.unit (trg f) \<star>\<^sub>S S.UP f)"
	using UP.lunit_coherence iso_lunit S.strict_lunit by simp
	thus ?thesis
	using UP.image_of_unitor(3) ide_f by presburger
	qed
	moreover have "(S.UP f \<star>\<^sub>S S.inv (UP.unit (src f))) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))
	= S.UP \<r>[f]"
	proof -
	have "S.UP \<r>[f] \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, src f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S UP.unit (src f)) = S.UP f"
	using UP.runit_coherence [of f] S.strict_runit by simp
	moreover have "S.iso (S.cmp\<^sub>U\<^sub>P (f, src f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S UP.unit (src f)))"
	using UP.unit_char UP.cmp_components_are_iso VV.arr_char S.UP_map\<^sub>0_obj
	by (intro S.isos_compose) auto
	ultimately have
	"S.UP \<r>[f] = S.UP f \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S UP.unit (src f)))"
	using S.invert_side_of_triangle(2)
	[of "S.UP f" "S.UP \<r>[f]" "S.cmp\<^sub>U\<^sub>P (f, src f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S UP.unit (src f))"]
	ideD(1) ide_f by blast
	thus ?thesis
	using ide_f UP.image_of_unitor(2) [of f] by argo
	qed
	ultimately show ?thesis
	using S.comp_assoc by simp
	qed
	also have "... = S.UP (\<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f])"
	by simp
	finally have "S.UP ((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>)) = S.UP (\<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f])"
	by simp
	moreover have "par ((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>)) (\<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f])"
	proof -
	have "\<guillemotleft>(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>"
	using E''.triangle_in_hom(1) by simp
	moreover have "\<guillemotleft>\<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f] : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>" by auto
	ultimately show ?thesis
	by (metis in_homE)
	qed
	ultimately show ?thesis
	using UP.is_faithful by blast
	qed
	thus "(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
	using E''.triangle_left_implies_right by simp
	qed
	show ?thesis
	using E''.adjoint_equivalence_in_bicategory_axioms E''.adjunction_in_bicategory_axioms
	adjunction_unit_determines_counit adjoint_equivalence_in_bicategory_def
	by metis
	qed

	lemma equivalence_map_extends_to_adjoint_equivalence:
	assumes "equivalence_map f"
	shows "\<exists>g \<eta> \<epsilon>. adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	proof -
	obtain g \<eta> \<epsilon>' where E: "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>'"
	using assms equivalence_map_def by auto
	interpret E: equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>'
	using E by auto
	obtain \<epsilon> where A: "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms equivalence_refines_to_adjoint_equivalence [of f g \<eta>]
	E.antipar E.unit_is_iso E.unit_in_hom by auto
	show ?thesis
	using E A by blast
	qed

	end

	subsection "Uniqueness of Adjoints"

	text \<open>
	Left and right adjoints determine each other up to isomorphism.
	\<close>

	context strict_bicategory
	begin

	lemma left_adjoint_determines_right_up_to_iso:
	assumes "adjoint_pair f g" and "adjoint_pair f g'"
	shows "g \<cong> g'"
	proof -
	obtain \<eta> \<epsilon> where A: "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms adjoint_pair_def by auto
	interpret A: adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using A by auto
	interpret A: adjunction_in_strict_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> ..
	obtain \<eta>' \<epsilon>' where A': "adjunction_in_bicategory V H \<a> \<i> src trg f g' \<eta>' \<epsilon>'"
	using assms adjoint_pair_def by auto
	interpret A': adjunction_in_bicategory V H \<a> \<i> src trg f g' \<eta>' \<epsilon>'
	using A' by auto
	interpret A': adjunction_in_strict_bicategory V H \<a> \<i> src trg f g' \<eta>' \<epsilon>' ..
	let ?\<phi> = "A'.trnl\<^sub>\<eta> g \<epsilon>"
	have "\<guillemotleft>?\<phi>: g \<Rightarrow> g'\<guillemotright>"
	using A'.trnl\<^sub>\<eta>_eq A'.adjoint_transpose_left(1) [of "trg f" g] A.antipar A'.antipar
	hcomp_arr_obj
	by auto
	moreover have "iso ?\<phi>"
	proof (intro isoI)
	let ?\<psi> = "A.trnl\<^sub>\<eta> g' \<epsilon>'"
	show "inverse_arrows ?\<phi> ?\<psi>"
	proof
	show "ide (?\<phi> \<cdot> ?\<psi>)"
	proof -
	have 1: "ide (trg f) \<and> trg (trg f) = trg f"
	by simp
	have "?\<phi> \<cdot> ?\<psi> = (g' \<star> \<epsilon>) \<cdot> ((\<eta>' \<star> g) \<cdot> (g \<star> \<epsilon>')) \<cdot> (\<eta> \<star> g')"
	using 1 A.antipar A'.antipar A.trnl\<^sub>\<eta>_eq [of "trg f" g' \<epsilon>']
	A'.trnl\<^sub>\<eta>_eq [of "trg f" g \<epsilon>] comp_assoc A.counit_in_hom A'.counit_in_hom
	by simp
	also have "... = ((g' \<star> \<epsilon>) \<cdot> (g' \<star> f \<star> g \<star> \<epsilon>')) \<cdot> ((\<eta>' \<star> g \<star> f \<star> g') \<cdot> (\<eta> \<star> g'))"
	proof -
	have "(\<eta>' \<star> g) \<cdot> (g \<star> \<epsilon>') = (\<eta>' \<star> g \<star> trg f) \<cdot> (src f \<star> g \<star> \<epsilon>')"
	using A.antipar A'.antipar hcomp_arr_obj hcomp_obj_arr [of "src f" "g \<star> \<epsilon>'"]
	hseqI'
	by (metis A'.counit_simps(1) A'.counit_simps(5) A.ide_right ideD(1)
	obj_trg trg_hcomp)
	also have "... = \<eta>' \<star> g \<star> \<epsilon>'"
	using A.antipar A'.antipar interchange [of \<eta>' "src f" "g \<star> trg f" "g \<star> \<epsilon>'"]
	whisker_left comp_arr_dom comp_cod_arr
	by simp
	also have "... = ((g' \<star> f) \<star> g \<star> \<epsilon>') \<cdot> (\<eta>' \<star> g \<star> (f \<star> g'))"
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar
	A'.unit_in_hom A'.counit_in_hom interchange whisker_left
	comp_arr_dom comp_cod_arr
	by (metis A'.counit_simps(1-2,5) A'.unit_simps(1,3) hseqI' ide_char)
	also have "... = (g' \<star> f \<star> g \<star> \<epsilon>') \<cdot> (\<eta>' \<star> g \<star> f \<star> g')"
	using hcomp_assoc by simp
	finally show ?thesis
	using comp_assoc by simp
	qed
	also have "... = (g' \<star> \<epsilon>') \<cdot> ((g' \<star> (\<epsilon> \<star> f) \<star> g') \<cdot> (g' \<star> (f \<star> \<eta>) \<star> g')) \<cdot> (\<eta>' \<star> g')"
	proof -
	have "(g' \<star> \<epsilon>) \<cdot> (g' \<star> f \<star> g \<star> \<epsilon>') = (g' \<star> \<epsilon>') \<cdot> (g' \<star> \<epsilon> \<star> f \<star> g')"
	proof -
	have "(g' \<star> \<epsilon>) \<cdot> (g' \<star> f \<star> g \<star> \<epsilon>') = g' \<star> \<epsilon> \<star> \<epsilon>'"
	proof -
	have "\<epsilon> \<cdot> (f \<star> g \<star> \<epsilon>') = \<epsilon> \<star> \<epsilon>'"
	using A.ide_left A.ide_right A.antipar A'.antipar hcomp_arr_obj comp_arr_dom
	comp_cod_arr interchange obj_src trg_src
	by (metis A'.counit_simps(1,3) A.counit_simps(1-2,4) hcomp_assoc)
	thus ?thesis
	using A.antipar A'.antipar whisker_left [of g' \<epsilon> "f \<star> g \<star> \<epsilon>'"]
	by (simp add: hcomp_assoc)
	qed
	also have "... = (g' \<star> \<epsilon>') \<cdot> (g' \<star> \<epsilon> \<star> f \<star> g')"
	proof -
	have "\<epsilon> \<star> \<epsilon>' = \<epsilon>' \<cdot> (\<epsilon> \<star> f \<star> g')"
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar
	hcomp_obj_arr hcomp_arr_obj comp_arr_dom comp_cod_arr interchange
	obj_src trg_src
	by (metis A'.counit_simps(1-2,5) A.counit_simps(1,3-4) arr_cod
	not_arr_null seq_if_composable)
	thus ?thesis
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar
	whisker_left
	by (metis A'.counit_simps(1,5) A.counit_simps(1,4) hseqI')
	qed
	finally show ?thesis by simp
	qed
	moreover have "(\<eta>' \<star> g \<star> f \<star> g') \<cdot> (\<eta> \<star> g') = (g' \<star> f \<star> \<eta> \<star> g') \<cdot> (\<eta>' \<star> g')"
	proof -
	have "(\<eta>' \<star> g \<star> f \<star> g') \<cdot> (\<eta> \<star> g') = \<eta>' \<star> \<eta> \<star> g'"
	proof -
	have "(\<eta>' \<star> g \<star> f) \<cdot> \<eta> = \<eta>' \<star> \<eta>"
	using A.ide_left A.ide_right A.antipar A'.antipar A'.unit_in_hom hcomp_arr_obj
	interchange comp_arr_dom comp_cod_arr
	by (metis A'.unit_simps(1-2,4) A.unit_simps(1,3,5) hcomp_obj_arr obj_trg)
	thus ?thesis
	using A.antipar A'.antipar whisker_right [of g' "\<eta>' \<star> g \<star> f" \<eta>]
	by (simp add: hcomp_assoc)
	qed
	also have "... = (g' \<star> f \<star> \<eta> \<star> g') \<cdot> (\<eta>' \<star> g')"
	proof -
	have "\<eta>' \<star> \<eta> = (g' \<star> f \<star> \<eta>) \<cdot> \<eta>'"
	using A.ide_left A.ide_right A.antipar A'.antipar A'.unit_in_hom hcomp_arr_obj
	comp_arr_dom comp_cod_arr hcomp_assoc interchange
	by (metis A'.unit_simps(1,3-4) A.unit_simps(1-2) obj_src)
	thus ?thesis
	using A.ide_left A.ide_right A.antipar A'.antipar A'.unit_in_hom hcomp_arr_obj
	whisker_right [of g' "g' \<star> f \<star> \<eta>" \<eta>']
	by (metis A'.ide_right A'.unit_simps(1,4) A.unit_simps(1,5)
	hseqI' hcomp_assoc)
	qed
	finally show ?thesis by simp
	qed
	ultimately show ?thesis
	using comp_assoc hcomp_assoc by simp
	qed
	also have "... = (g' \<star> \<epsilon>') \<cdot> ((g' \<star> f) \<star> g') \<cdot> (\<eta>' \<star> g')"
	proof -
	have "(g' \<star> (\<epsilon> \<star> f) \<star> g') \<cdot> (g' \<star> (f \<star> \<eta>) \<star> g') = g' \<star> f \<star> g'"
	proof -
	have "(g' \<star> (\<epsilon> \<star> f) \<star> g') \<cdot> (g' \<star> (f \<star> \<eta>) \<star> g') =
	g' \<star> ((\<epsilon> \<star> f) \<star> g') \<cdot> ((f \<star> \<eta>) \<star> g')"
	using A.ide_left A.ide_right A.antipar A'.antipar A'.unit_in_hom
	A'.counit_in_hom whisker_left [of g' "(\<epsilon> \<star> f) \<star> g'" "(f \<star> \<eta>) \<star> g'"]
	by (metis A'.ide_right A.triangle_left hseqI' ideD(1) whisker_right)
	also have "... = g' \<star> (\<epsilon> \<star> f) \<cdot> (f \<star> \<eta>) \<star> g'"
	using A.antipar A'.antipar whisker_right [of g' "\<epsilon> \<star> f" "f \<star> \<eta>"]
	by (simp add: A.triangle_left)
	also have "... = g' \<star> f \<star> g'"
	using A.triangle_left by simp
	finally show ?thesis by simp
	qed
	thus ?thesis
	using hcomp_assoc by simp
	qed
	also have "... = (g' \<star> \<epsilon>') \<cdot> (\<eta>' \<star> g')"
	using A.antipar A'.antipar A'.unit_in_hom A'.counit_in_hom comp_cod_arr
	by (metis A'.ide_right A'.triangle_in_hom(2) A.ide_left arrI assoc_is_natural_2
	ide_char seqE strict_assoc)
	also have "... = g'"
	using A'.triangle_right by simp
	finally have "?\<phi> \<cdot> ?\<psi> = g'" by simp
	thus ?thesis by simp
	qed
	show "ide (?\<psi> \<cdot> ?\<phi>)"
	proof -
	have 1: "ide (trg f) \<and> trg (trg f) = trg f"
	by simp
	have "?\<psi> \<cdot> ?\<phi> = (g \<star> \<epsilon>') \<cdot> ((\<eta> \<star> g') \<cdot> (g' \<star> \<epsilon>)) \<cdot> (\<eta>' \<star> g)"
	using A.antipar A'.antipar A'.trnl\<^sub>\<eta>_eq [of "trg f" g \<epsilon>]
	A.trnl\<^sub>\<eta>_eq [of "trg f" g' \<epsilon>'] comp_assoc A.counit_in_hom A'.counit_in_hom
	by simp
	also have "... = ((g \<star> \<epsilon>') \<cdot> (g \<star> f \<star> g' \<star> \<epsilon>)) \<cdot> ((\<eta> \<star> g' \<star> f \<star> g) \<cdot> (\<eta>' \<star> g))"
	proof -
	have "(\<eta> \<star> g') \<cdot> (g' \<star> \<epsilon>) = (\<eta> \<star> g' \<star> trg f) \<cdot> (src f \<star> g' \<star> \<epsilon>)"
	using A.antipar A'.antipar hcomp_arr_obj hcomp_obj_arr hseqI'
	by (metis A'.ide_right A.unit_simps(1,4) hcomp_assoc hcomp_obj_arr
	ideD(1) obj_src)
	also have "... = \<eta> \<star> g' \<star> \<epsilon>"
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar A.unit_in_hom
	A.counit_in_hom interchange
	by (metis "1" A.counit_simps(5) A.unit_simps(4) hseqI' ide_def ide_in_hom(2)
	not_arr_null seqI' src.preserves_ide)
	also have "... = ((g \<star> f) \<star> g' \<star> \<epsilon>) \<cdot> (\<eta> \<star> g' \<star> (f \<star> g))"
	using A'.ide_right A'.antipar interchange ide_char comp_arr_dom comp_cod_arr hseqI'
	by (metis A.counit_simps(1-2,5) A.unit_simps(1,3))
	also have "... = (g \<star> f \<star> g' \<star> \<epsilon>) \<cdot> (\<eta> \<star> g' \<star> f \<star> g)"
	using hcomp_assoc by simp
	finally show ?thesis
	using comp_assoc by simp
	qed
	also have "... = (g \<star> \<epsilon>) \<cdot> ((g \<star> (\<epsilon>' \<star> f) \<star> g) \<cdot> (g \<star> (f \<star> \<eta>') \<star> g)) \<cdot> (\<eta> \<star> g)"
	proof -
	have "(g \<star> \<epsilon>') \<cdot> (g \<star> f \<star> g' \<star> \<epsilon>) = (g \<star> \<epsilon>) \<cdot> (g \<star> \<epsilon>' \<star> f \<star> g)"
	proof -
	have "(g \<star> \<epsilon>') \<cdot> (g \<star> f \<star> g' \<star> \<epsilon>) = g \<star> \<epsilon>' \<star> \<epsilon>"
	proof -
	have "\<epsilon>' \<cdot> (f \<star> g' \<star> \<epsilon>) = \<epsilon>' \<star> \<epsilon>"
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar hcomp_arr_obj
	comp_arr_dom comp_cod_arr interchange obj_src trg_src hcomp_assoc
	by (metis A.counit_simps(1,3) A'.counit_simps(1-2,4))
	thus ?thesis
	using A.antipar A'.antipar whisker_left [of g \<epsilon>' "f \<star> g' \<star> \<epsilon>"]
	by (simp add: hcomp_assoc)
	qed
	also have "... = (g \<star> \<epsilon>) \<cdot> (g \<star> \<epsilon>' \<star> f \<star> g)"
	proof -
	have "\<epsilon>' \<star> \<epsilon> = \<epsilon> \<cdot> (\<epsilon>' \<star> f \<star> g)"
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar hcomp_obj_arr
	hcomp_arr_obj comp_arr_dom comp_cod_arr interchange obj_src trg_src
	by (metis A.counit_simps(1-2,5) A'.counit_simps(1,3-4)
	arr_cod not_arr_null seq_if_composable)
	thus ?thesis
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar
	whisker_left
	by (metis A.counit_simps(1,5) A'.counit_simps(1,4) hseqI')
	qed
	finally show ?thesis by simp
	qed
	moreover have "(\<eta> \<star> g' \<star> f \<star> g) \<cdot> (\<eta>' \<star> g) = (g \<star> f \<star> \<eta>' \<star> g) \<cdot> (\<eta> \<star> g)"
	proof -
	have "(\<eta> \<star> g' \<star> f \<star> g) \<cdot> (\<eta>' \<star> g) = \<eta> \<star> \<eta>' \<star> g"
	proof -
	have "(\<eta> \<star> g' \<star> f) \<cdot> \<eta>' = \<eta> \<star> \<eta>'"
	using A.antipar A'.antipar A.unit_in_hom hcomp_arr_obj
	comp_arr_dom comp_cod_arr hcomp_obj_arr interchange
	by (metis A'.unit_simps(1,3,5) A.unit_simps(1-2,4) obj_trg)
	thus ?thesis
	using A.antipar A'.antipar whisker_right [of g "\<eta> \<star> g' \<star> f" \<eta>']
	by (simp add: hcomp_assoc)
	qed
	also have "... = ((g \<star> f) \<star> \<eta>' \<star> g) \<cdot> (\<eta> \<star> src f \<star> g)"
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar A.unit_in_hom
	A'.unit_in_hom comp_arr_dom comp_cod_arr interchange
	by (metis A'.unit_simps(1-2,4) A.unit_simps(1,3) hseqI' ide_char)
	also have "... = (g \<star> f \<star> \<eta>' \<star> g) \<cdot> (\<eta> \<star> g)"
	using A.antipar A'.antipar hcomp_assoc
	by (simp add: hcomp_obj_arr)
	finally show ?thesis by simp
	qed
	ultimately show ?thesis
	using comp_assoc hcomp_assoc by simp
	qed
	also have "... = (g \<star> \<epsilon>) \<cdot> ((g \<star> f) \<star> g) \<cdot> (\<eta> \<star> g)"
	proof -
	have "(g \<star> (\<epsilon>' \<star> f) \<star> g) \<cdot> (g \<star> (f \<star> \<eta>') \<star> g) = g \<star> f \<star> g"
	proof -
	have "(g \<star> (\<epsilon>' \<star> f) \<star> g) \<cdot> (g \<star> (f \<star> \<eta>') \<star> g) =
	g \<star> ((\<epsilon>' \<star> f) \<star> g) \<cdot> ((f \<star> \<eta>') \<star> g)"
	using A.ide_left A.ide_right A'.ide_right A.antipar A'.antipar A.unit_in_hom
	A.counit_in_hom whisker_left
	by (metis A'.triangle_left hseqI' ideD(1) whisker_right)
	also have "... = g \<star> (\<epsilon>' \<star> f) \<cdot> (f \<star> \<eta>') \<star> g"
	using A.antipar A'.antipar whisker_right [of g "\<epsilon>' \<star> f" "f \<star> \<eta>'"]
	by (simp add: A'.triangle_left)
	also have "... = g \<star> f \<star> g"
	using A'.triangle_left by simp
	finally show ?thesis by simp
	qed
	thus ?thesis
	using hcomp_assoc by simp
	qed
	also have "... = (g \<star> \<epsilon>) \<cdot> (\<eta> \<star> g)"
	using A.antipar A'.antipar A.unit_in_hom A.counit_in_hom comp_cod_arr
	by (metis A.ide_left A.ide_right A.triangle_in_hom(2) arrI assoc_is_natural_2
	ide_char seqE strict_assoc)
	also have "... = g"
	using A.triangle_right by simp
	finally have "?\<psi> \<cdot> ?\<phi> = g" by simp
	moreover have "ide g"
	by simp
	ultimately show ?thesis by simp
	qed
	qed
	qed
	ultimately show ?thesis
	using isomorphic_def by auto
	qed

	end

	text \<open>
	We now use strictification to extend to arbitrary bicategories.
	\<close>

	context bicategory
	begin

	interpretation S: strictified_bicategory V H \<a> \<i> src trg ..

	notation S.vcomp (infixr "\<cdot>\<^sub>S" 55)
	notation S.hcomp (infixr "\<star>\<^sub>S" 53)
	notation S.in_hom ("\<guillemotleft>_ : _ \<Rightarrow>\<^sub>S _\<guillemotright>")
	notation S.in_hhom ("\<guillemotleft>_ : _ \<rightarrow>\<^sub>S _\<guillemotright>")

	interpretation UP: equivalence_pseudofunctor V H \<a> \<i> src trg
	S.vcomp S.hcomp S.\<a> S.\<i> S.src S.trg S.UP S.cmp\<^sub>U\<^sub>P
	using S.UP_is_equivalence_pseudofunctor by auto
	interpretation UP: pseudofunctor_into_strict_bicategory V H \<a> \<i> src trg
	S.vcomp S.hcomp S.\<a> S.\<i> S.src S.trg S.UP S.cmp\<^sub>U\<^sub>P
	..
	interpretation UP: fully_faithful_functor V S.vcomp S.UP
	using S.UP_is_fully_faithful_functor by auto

	lemma left_adjoint_determines_right_up_to_iso:
	assumes "adjoint_pair f g" and "adjoint_pair f g'"
	shows "g \<cong> g'"
	proof -
	have 0: "ide g \<and> ide g'"
	using assms adjoint_pair_def adjunction_in_bicategory_def
	adjunction_data_in_bicategory_def adjunction_data_in_bicategory_axioms_def
	by metis
	have 1: "S.adjoint_pair (S.UP f) (S.UP g) \<and> S.adjoint_pair (S.UP f) (S.UP g')"
	using assms UP.preserves_adjoint_pair by simp
	obtain \<nu> where \<nu>: "\<guillemotleft>\<nu> : S.UP g \<Rightarrow>\<^sub>S S.UP g'\<guillemotright> \<and> S.iso \<nu>"
	using 1 S.left_adjoint_determines_right_up_to_iso S.isomorphic_def by blast
	obtain \<mu> where \<mu>: "\<guillemotleft>\<mu> : g \<Rightarrow> g'\<guillemotright> \<and> S.UP \<mu> = \<nu>"
	using 0 \<nu> UP.is_full [of g' g \<nu>] by auto
	have "\<guillemotleft>\<mu> : g \<Rightarrow> g'\<guillemotright> \<and> iso \<mu>"
	using \<mu> \<nu> UP.reflects_iso by auto
	thus ?thesis
	using isomorphic_def by auto
	qed

	lemma right_adjoint_determines_left_up_to_iso:
	assumes "adjoint_pair f g" and "adjoint_pair f' g"
	shows "f \<cong> f'"
	proof -
	obtain \<eta> \<epsilon> where A: "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms adjoint_pair_def by auto
	interpret A: adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using A by auto
	obtain \<eta>' \<epsilon>' where A': "adjunction_in_bicategory V H \<a> \<i> src trg f' g \<eta>' \<epsilon>'"
	using assms adjoint_pair_def by auto
	interpret A': adjunction_in_bicategory V H \<a> \<i> src trg f' g \<eta>' \<epsilon>'
	using A' by auto
	interpret Cop: op_bicategory V H \<a> \<i> src trg ..
	interpret Aop: adjunction_in_bicategory V Cop.H Cop.\<a> \<i> Cop.src Cop.trg g f \<eta> \<epsilon>
	using A.antipar A.triangle_left A.triangle_right Cop.assoc_ide_simp
	Cop.lunit_ide_simp Cop.runit_ide_simp
	by (unfold_locales, auto)
	interpret Aop': adjunction_in_bicategory V Cop.H Cop.\<a> \<i> Cop.src Cop.trg g f' \<eta>' \<epsilon>'
	using A'.antipar A'.triangle_left A'.triangle_right Cop.assoc_ide_simp
	Cop.lunit_ide_simp Cop.runit_ide_simp
	by (unfold_locales, auto)
	show ?thesis
	using Aop.adjunction_in_bicategory_axioms Aop'.adjunction_in_bicategory_axioms
	Cop.left_adjoint_determines_right_up_to_iso Cop.adjoint_pair_def
	by blast
	qed

	end

	context chosen_right_adjoints
	begin

	lemma isomorphic_to_left_adjoint_implies_isomorphic_right_adjoint:
	assumes "is_left_adjoint f" and "f \<cong> h"
	shows "f\<^sup>* \<cong> h\<^sup>*"
	proof -
	have 1: "adjoint_pair f f\<^sup>*"
	using assms left_adjoint_extends_to_adjoint_pair by blast
	moreover have "adjoint_pair h f\<^sup>*"
	using assms 1 adjoint_pair_preserved_by_iso isomorphic_symmetric isomorphic_reflexive
	by (meson isomorphic_def right_adjoint_simps(1))
	thus ?thesis
	using left_adjoint_determines_right_up_to_iso left_adjoint_extends_to_adjoint_pair
	by blast
	qed

	end

	context bicategory
	begin

	lemma equivalence_is_adjoint:
	assumes "equivalence_map f"
	shows equivalence_is_left_adjoint: "is_left_adjoint f"
	and equivalence_is_right_adjoint: "is_right_adjoint f"
	proof -
	obtain g \<eta> \<epsilon> where fg: "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms equivalence_map_extends_to_adjoint_equivalence by blast
	interpret fg: adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using fg by simp
	interpret gf: adjoint_equivalence_in_bicategory V H \<a> \<i> src trg g f \<open>inv \<epsilon>\<close> \<open>inv \<eta>\<close>
	using fg.dual_adjoint_equivalence by simp
	show "is_left_adjoint f"
	using fg.adjunction_in_bicategory_axioms adjoint_pair_def by auto
	show "is_right_adjoint f"
	using gf.adjunction_in_bicategory_axioms adjoint_pair_def by auto
	qed

	lemma right_adjoint_to_equivalence_is_equivalence:
	assumes "equivalence_map f" and "adjoint_pair f g"
	shows "equivalence_map g"
	proof -
	obtain \<eta> \<epsilon> where fg: "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms adjoint_pair_def by auto
	interpret fg: adjunction_in_bicategory \<open>(\<cdot>)\<close> \<open>(\<star>)\<close> \<a> \<i> src trg f g \<eta> \<epsilon>
	using fg by simp
	obtain g' \<phi> \<psi> where fg': "equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g' \<phi> \<psi>"
	using assms equivalence_map_def by auto
	interpret fg': equivalence_in_bicategory \<open>(\<cdot>)\<close> \<open>(\<star>)\<close> \<a> \<i> src trg f g' \<phi> \<psi>
	using fg' by auto
	obtain \<psi>' where \<psi>': "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g' \<phi> \<psi>'"
	using assms equivalence_refines_to_adjoint_equivalence [of f g' \<phi>]
	fg'.antipar fg'.unit_in_hom fg'.unit_is_iso
	by auto
	interpret \<psi>': adjoint_equivalence_in_bicategory \<open>(\<cdot>)\<close> \<open>(\<star>)\<close> \<a> \<i> src trg f g' \<phi> \<psi>'
	using \<psi>' by simp
	have 1: "g \<cong> g'"
	using fg.adjunction_in_bicategory_axioms \<psi>'.adjunction_in_bicategory_axioms
	left_adjoint_determines_right_up_to_iso adjoint_pair_def
	by blast
	obtain \<gamma> where \<gamma>: "\<guillemotleft>\<gamma> : g' \<Rightarrow> g\<guillemotright> \<and> iso \<gamma>"
	using 1 isomorphic_def isomorphic_symmetric by metis
	have "equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g ((\<gamma> \<star> f) \<cdot> \<phi>) (\<psi>' \<cdot> (f \<star> inv \<gamma>))"
	using \<gamma> equivalence_preserved_by_iso_right \<psi>'.equivalence_in_bicategory_axioms by simp
	hence "quasi_inverses f g"
	using quasi_inverses_def by blast
	thus ?thesis
	using equivalence_mapI quasi_inverses_symmetric by blast
	qed

	lemma left_adjoint_to_equivalence_is_equivalence:
	assumes "equivalence_map f" and "adjoint_pair g f"
	shows "equivalence_map g"
	proof -
	obtain \<eta> \<epsilon> where gf: "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g f \<eta> \<epsilon>"
	using assms adjoint_pair_def by auto
	interpret gf: adjunction_in_bicategory \<open>(\<cdot>)\<close> \<open>(\<star>)\<close> \<a> \<i> src trg g f \<eta> \<epsilon>
	using gf by simp
	obtain g' where 1: "quasi_inverses g' f"
	using assms equivalence_mapE quasi_inverses_symmetric by blast
	obtain \<phi> \<psi> where g'f: "equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g' f \<phi> \<psi>"
	using assms 1 quasi_inverses_def by auto
	interpret g'f: equivalence_in_bicategory \<open>(\<cdot>)\<close> \<open>(\<star>)\<close> \<a> \<i> src trg g' f \<phi> \<psi>
	using g'f by auto
	obtain \<psi>' where \<psi>': "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g' f \<phi> \<psi>'"
	using assms 1 equivalence_refines_to_adjoint_equivalence [of g' f \<phi>]
	g'f.antipar g'f.unit_in_hom g'f.unit_is_iso quasi_inverses_def
	equivalence_map_def
	by auto
	interpret \<psi>': adjoint_equivalence_in_bicategory \<open>(\<cdot>)\<close> \<open>(\<star>)\<close> \<a> \<i> src trg g' f \<phi> \<psi>'
	using \<psi>' by simp
	have 1: "g \<cong> g'"
	using gf.adjunction_in_bicategory_axioms \<psi>'.adjunction_in_bicategory_axioms
	right_adjoint_determines_left_up_to_iso adjoint_pair_def
	by blast
	obtain \<gamma> where \<gamma>: "\<guillemotleft>\<gamma> : g' \<Rightarrow> g\<guillemotright> \<and> iso \<gamma>"
	using 1 isomorphic_def isomorphic_symmetric by metis
	have "equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g f ((f \<star> \<gamma>) \<cdot> \<phi>) (\<psi>' \<cdot> (inv \<gamma> \<star> f))"
	using \<gamma> equivalence_preserved_by_iso_left \<psi>'.equivalence_in_bicategory_axioms by simp
	hence "quasi_inverses g f"
	using quasi_inverses_def by auto
	thus ?thesis
	using quasi_inverses_symmetric quasi_inverses_def equivalence_map_def by blast
	qed

	lemma quasi_inverses_are_adjoint_pair:
	assumes "quasi_inverses f g"
	shows "adjoint_pair f g"
	proof -
	obtain \<eta> \<epsilon> where \<eta>\<epsilon>: "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
	using assms quasi_inverses_def by auto
	interpret \<eta>\<epsilon>: equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>
	using \<eta>\<epsilon> by auto
	obtain \<epsilon>' where \<eta>\<epsilon>': "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>'"
	using \<eta>\<epsilon> equivalence_map_def \<eta>\<epsilon>.antipar \<eta>\<epsilon>.unit_in_hom \<eta>\<epsilon>.unit_is_iso
	\<eta>\<epsilon>.ide_right equivalence_refines_to_adjoint_equivalence [of f g \<eta>]
	by force
	interpret \<eta>\<epsilon>': adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>'
	using \<eta>\<epsilon>' by auto
	show ?thesis
	using \<eta>\<epsilon>' adjoint_pair_def \<eta>\<epsilon>'.adjunction_in_bicategory_axioms by auto
	qed

	lemma quasi_inverses_isomorphic_right:
	assumes "quasi_inverses f g"
	shows "quasi_inverses f g' \<longleftrightarrow> g \<cong> g'"
	proof
	show "g \<cong> g' \<Longrightarrow> quasi_inverses f g'"
	using assms quasi_inverses_def isomorphic_def equivalence_preserved_by_iso_right
	by metis
	assume g': "quasi_inverses f g'"
	show "g \<cong> g'"
	using assms g' quasi_inverses_are_adjoint_pair left_adjoint_determines_right_up_to_iso
	by blast
	qed

	lemma quasi_inverses_isomorphic_left:
	assumes "quasi_inverses f g"
	shows "quasi_inverses f' g \<longleftrightarrow> f \<cong> f'"
	proof
	show "f \<cong> f' \<Longrightarrow> quasi_inverses f' g"
	using assms quasi_inverses_def isomorphic_def equivalence_preserved_by_iso_left
	by metis
	assume f': "quasi_inverses f' g"
	show "f \<cong> f'"
	using assms f' quasi_inverses_are_adjoint_pair right_adjoint_determines_left_up_to_iso
	by blast
	qed

	end

	end