Lemma homset_conj'_natural_precomp {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 ;; ff) = transportb _ (φ_adj_natural_precomp A _ _ f _ k) (kk ;; homset_conj' _ _ _ ff). Proof. unfold homset_conj'. rewrite disp_functor_comp. unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite 2 assoc_disp. cbn. set (nat_ηη := disp_nat_trans_ax_var ηη kk). cbn in nat_ηη. rewrite nat_ηη. unfold transportb. rewrite mor_disp_transportf_postwhisker. 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.