Lemma BinCoproductArrowUnique (a b : C) (CC : BinCoproduct a b) (x : C) (f : a --> x) (g : b --> x) (k : BinCoproductObject CC --> x) : BinCoproductIn1 CC · k = f → BinCoproductIn2 CC · k = g → k = BinCoproductArrow CC f g. Proof. intros H1 H2. set (H := tpair (λ h, dirprod _ _ ) k (make_dirprod H1 H2)). set (H' := (pr2 (isBinCoproduct_BinCoproduct CC _ f g)) H). apply (base_paths _ _ H'). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. 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.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.ProductCategory.