Proposition pr1_idtoiso_disp_functor {C₁ C₂ : category} {F : C₁ ⟶ C₂} {D₁ : disp_cat C₁} {D₂ : disp_cat C₂} (FF : disp_functor F D₁ D₂) {x : C₁} {xx yy : D₁ x} (p : xx = yy) : pr1 (idtoiso_disp (idpath (F x)) (maponpaths (FF x) p)) = transportf (λ z, _ -->[ z ] _) (functor_id F _) (♯FF (idtoiso_disp (idpath x) p)). Proof. induction p. cbn. rewrite disp_functor_id. rewrite transportfbinv. apply idpath. 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.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos.