Lemma is_nat_trans_coproduct_nat_trans_data : is_nat_trans _ _ coproduct_nat_trans_data. Proof. intros a b k; simpl. eapply pathscomp0. apply (precompWithCoproductArrow I D (HD (λ i : I, (F i) a)) (HD (λ i : I, (F i) b))). apply pathsinv0. eapply pathscomp0; [apply postcompWithCoproductArrow|]. apply maponpaths, funextsec; intro i. now rewrite (nat_trans_ax (f i)). 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.