Lemma disp_nat_trans_id_right {C' C : category} {D' : disp_cat C'} {D : disp_cat C} (F' F : functor C' C) (a : nat_trans F' F) (FF' : disp_functor F' D' D) (FF : disp_functor F D' D) (b : disp_nat_trans a FF' FF) : disp_nat_trans_comp b (disp_nat_trans_id FF) = transportb (λ f : nat_trans F' F, disp_nat_trans f FF' FF) (id_right (a : [C',C] ⟦ _ , _ ⟧)) b. 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 id_right_disp. unfold transportb. apply maponpaths_2, 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.