Lemma homset_conj_inv_after_conj' {c : C} {c' : C'} (f : C'⟦F c, c'⟧)(d : D c) (d' : D' c') (beta : FF _ d -->[f] d') : transportf _ (φ_adj_inv_after_φ_adj A f) (homset_conj_inv _ _ _ (homset_conj' f d d' beta)) = beta. Proof. unfold homset_conj'. cbn. set (eq := homset_conj_inv_natural_postcomp (ηη c d) beta). cbn in eq. rewrite eq. unfold homset_conj_inv. cbn. rewrite transport_f_b. assert (triangle1 : ♯ FF (ηη c d);; εε (F c) (FF c d) = transportb _ (triangle_id_left_ad A c ) (id_disp _)) by (exact ((pr1 (pr2 X)) c d)). cbn in triangle1. rewrite triangle1. unfold transportb. rewrite mor_disp_transportf_postwhisker. rewrite id_left_disp. unfold transportb. rewrite 2 transport_f_f. intermediate_path (transportf _ (idpath _) beta). - apply maponpaths_2, homset_property. - apply idpath. Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences.