Lemma homset_conj'_natural_postcomp {c : C} {c' : C'} {f : C'⟦F c, c'⟧} {c'' : C'} {k : C'⟦c', c''⟧} {d : D c} {d' : D' c'} {d'' : D' c''} (ff : FF _ d -->[f] d') (kk : d' -->[k] d'') : homset_conj' _ _ _ (ff ;; kk) = transportb _ (φ_adj_natural_postcomp A _ _ f _ k) (homset_conj' _ _ _ ff ;; ♯ GG kk). Proof. unfold homset_conj'. rewrite disp_functor_comp. rewrite assoc_disp_var. unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite 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.