Lemma coproduct_of_functors_nat_trans_aux {I : UU} {C D : category} (CP : Coproducts I D) {F G : I → C ⟶ D} (α : ∏ i : I, nat_trans (F i) (G i)) : is_nat_trans (coproduct_of_functors I C D CP F) (coproduct_of_functors I C D CP G) (λ c : C, CoproductOfArrows I D (CP (λ i : I, F i c)) (CP (λ i : I, G i c)) (λ i : I, α i c)). Proof. intros c1 c2 f. etrans. 1: apply precompWithCoproductArrow. etrans. 2: apply pathsinv0, precompWithCoproductArrow. apply maponpaths. apply funextsec ; intro i. etrans. 1: apply assoc. etrans. 2: apply assoc'. apply maponpaths_2. exact (pr2 (α i) _ _ f). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Limits. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All.