Definition total_functor_identity {C : category} (D : disp_cat C) : functor_identity (total_category D) ⟹ total_functor (disp_functor_identity D). Proof. use make_nat_trans. - exact (λ _, identity _). - abstract (intros ? ? ? ; simpl ; refine (@id_right (total_category _) _ _ _ @ _) ; exact (!(@id_left (total_category _) _ _ _))). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.AxiomOfChoice. 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.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence.