Definition isBinCoproduct_eq_arrow {C : category} {x y z : C} {ι₁ ι₁' : x --> z} (p₁ : ι₁ = ι₁') {ι₂ ι₂' : y --> z} (p₂ : ι₂ = ι₂') (H : isBinCoproduct C x y z ι₁ ι₂) : isBinCoproduct C x y z ι₁' ι₂'. Proof. pose (P := make_BinCoproduct _ _ _ _ _ _ H). intros w f g. use iscontraprop1. - abstract (induction p₁, p₂ ; apply isapropifcontr ; apply H). - simple refine (_ ,, _ ,, _). + exact (BinCoproductArrow P f g). + abstract (induction p₁ ; exact (BinCoproductIn1Commutes _ _ _ P _ f g)). + abstract (induction p₂ ; exact (BinCoproductIn2Commutes _ _ _ P _ f g)). Defined.
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.