Definition composite_disp_cat_id_comp {C : category} (DD : disp_disp_cat C) : disp_cat_id_comp C (composite_disp_cat_ob_mor DD). Proof. destruct DD as [D E]. use tpair. - intros x [xx xxx]. use tpair. + simpl. exact (id_disp xx). + simpl. exact (id_disp xxx). - simpl. intros x y z f g [xx xxx] [yy yyy] [zz zzz] [ff fff] [gg ggg]. use tpair. + exact (ff ;; gg). + simpl. exact (fff ;; ggg). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Foundations.All.