Lemma disp_nat_trans_assoc {C' C : category} {D' : disp_cat C'} {D : disp_cat C} (x y z w : functor C' C) (f : nat_trans x y) (g : nat_trans y z) (h : nat_trans z w) (xx : disp_functor x D' D) (yy : disp_functor y D' D) (zz : disp_functor z D' D) (ww : disp_functor w D' D) (ff : disp_nat_trans f xx yy) (gg : disp_nat_trans g yy zz) (hh : disp_nat_trans h zz ww) : disp_nat_trans_comp ff (disp_nat_trans_comp gg hh) = transportb (λ f0 : nat_trans x w, disp_nat_trans f0 xx ww) (assoc (f : [C', C] ⟦_,_⟧) g h) (disp_nat_trans_comp (disp_nat_trans_comp ff gg) hh). Proof. apply subtypePath. { intro. apply isaprop_disp_nat_trans_axioms. } apply funextsec; intro c'. apply funextsec; intro xx'. apply pathsinv0. etrans. apply disp_nat_trans_transportf. apply pathsinv0. etrans. apply assoc_disp. unfold transportb. apply maponpaths_2. apply homset_property. Qed.
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.Core.Isos. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Isos.