Lemma CoproductArrowUnique (a b : C) (CC : CoproductCocone a b) (x : C) (f : a --> x) (g : b --> x) (k : CoproductObject CC --> x) : CoproductIn1 CC · k = f -> CoproductIn2 CC · k = g -> k = CoproductArrow CC f g. Proof. intros u v. apply binarySumMapUniqueness. { refine (u @ _). apply pathsinv0, CoproductIn1Commutes. } { refine (v @ _). apply pathsinv0, CoproductIn2Commutes. } Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.RepresentableFunctors.Representation.