Lemma Coproduct_endo_is_identity (a : I -> C) (CC : Coproduct a) (k : CoproductObject CC --> CoproductObject CC) (H1 : ∏ i, CoproductIn CC i · k = CoproductIn CC i) : identity _ = k. Proof. apply pathsinv0. eapply pathscomp0; [apply CoproductArrowEta|]. apply pathsinv0, CoproductArrowUnique; intro i; apply pathsinv0. now rewrite id_right, H1. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.ProductCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.whiskering.