Lemma precompWithCoproductArrow {a b : C} (CCab : CoproductCocone a b) {c d : C} (CCcd : CoproductCocone c d) (f : a --> c) (g : b --> d) {x : C} (k : c --> x) (h : d --> x) : CoproductOfArrows CCab CCcd f g · CoproductArrow CCcd k h = CoproductArrow CCab (f · k) (g · h). Proof. apply CoproductArrowUnique. - rewrite assoc. rewrite CoproductOfArrowsIn1. rewrite <- assoc, CoproductIn1Commutes. apply idpath. - rewrite assoc, CoproductOfArrowsIn2. rewrite <- assoc, CoproductIn2Commutes. apply idpath. 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.