Lemma homset_conj_inv_natural_postcomp {c : C} {c' : C'} {g : C⟦c, G c'⟧} {c'' : C'} {f : C'⟦c', c''⟧} {d : D c} {d' : D' c'} {d'' : D' c''} (gg : d -->[g] GG _ d') (ff : d' -->[f] d'') : homset_conj_inv _ _ _ (gg ;; ♯ GG ff) = transportb _ (φ_adj_inv_natural_postcomp A _ _ g _ f) (homset_conj_inv _ _ _ gg ;; ff). Proof. unfold homset_conj_inv. rewrite disp_functor_comp. unfold transportb. rewrite mor_disp_transportf_postwhisker. rewrite 2 assoc_disp_var. cbn. set (nat_εε := disp_nat_trans_ax εε ff). cbn in nat_εε. rewrite nat_εε. unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite 3 transport_f_f. apply maponpaths_2, homset_property. 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.