Definition disp_cofunctormaps_cat_id_comp : disp_cat_id_comp _ disp_cofunctormaps_cat_ob_mor. Proof. apply tpair. - intros (C, (ty, tm)) p. apply nat_trans_eq. + apply homset_property. + cbn. intros. etrans. * apply id_right. * apply pathsinv0. apply id_left. - intros (C, (ty, tm)). intros (C', (ty', tm')). intros (C'', (ty'', tm'')). cbn in *. intros (f, (αty,αtm)). intros (g, (βty,βtm)). cbn in *. intros p p' p''. cbn in *. intros eq1 eq2. apply nat_trans_eq. + apply homset_property. + cbn. intros x. set (h1 := nat_trans_eq_pointwise eq1 x). set (h2 := nat_trans_eq_pointwise eq2 (f x)). cbn in *. rewrite assoc. rewrite h1. rewrite <- assoc. rewrite <- assoc. apply maponpaths. apply h2. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DisplayedBicats.Examples.ContravariantFunctor. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sigma.