Lemma precompWithCoproductArrowInclusion {I J : UU} {C : category} (incl : I -> J) {a : I -> C} (CCab : Coproduct _ _ a) {c : J -> C} (CCcd : Coproduct _ _ c) (f : ∏ i, a i --> c (incl i)) {x : C} (k : ∏ i, c i --> x) : CoproductOfArrowsInclusion incl CCab CCcd f · CoproductArrow _ _ CCcd k = CoproductArrow _ _ CCab (λ i, f i · k (incl i)). Proof. apply CoproductArrowUnique; intro i. now rewrite assoc, CoproductOfArrowsInclusionIn, <- assoc, CoproductInCommutes. 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.