Definition iso_to_Initial {C : category} (I : Initial C) (x : C) (i : z_iso I x) : isInitial C x. Proof. intros w. use iscontraprop1. - abstract (use invproofirrelevance ; intros φ₁ φ₂ ; refine (!(id_left _) @ _ @ id_left _) ; rewrite <- !(z_iso_after_z_iso_inv i) ; rewrite !assoc' ; apply maponpaths ; apply InitialArrowEq). - exact (z_iso_inv i · InitialArrow I w). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. 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.FunctorCategory. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Limits.Coproducts.