Proposition is_functor_to_iterated_slice_data : is_functor to_iterated_slice_data. Proof. split. - intros h. use eq_mor_cod_fib. use eq_mor_cod_fib ; cbn. apply idpath. - intros h₁ h₂ h₃ p q. use eq_mor_cod_fib. use eq_mor_cod_fib. rewrite comp_in_cod_fib. etrans. { simpl. apply comp_in_cod_fib. } refine (!_). etrans. { apply comp_in_cod_fib. } cbn. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.PreservationProperties. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodColimits.