Definition to_initial_slice {C : category} {y : C} (I : C/y) (H : isInitial C (cod_dom I)) : isInitial (C/y) I. Proof. intros f. pose (II := make_Initial _ H). use iscontraprop1. - abstract (use invproofirrelevance ; intros φ₁ φ₂ ; use eq_mor_cod_fib ; use (InitialArrowEq (O := II))). - use make_cod_fib_mor. + exact (InitialArrow II _). + abstract (use (InitialArrowEq (O := II))). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod.