Lemma is_nat_trans_coproduct_nat_trans_data : is_nat_trans _ _ coproduct_nat_trans_data. Proof. intros a b k. simpl. unfold BinCoproduct_of_functors_mor. unfold coproduct_nat_trans_data. simpl. set (XX:=precompWithBinCoproductArrow). set (X1 := XX D _ _ (HD (F a) (G a))). set (X2 := X1 _ _ (HD (F b) (G b))). rewrite X2. clear X2 X1 XX. set (XX:=postcompWithBinCoproductArrow). set (X1 := XX D _ _ (HD (F a) (G a))). rewrite X1. rewrite (nat_trans_ax f). rewrite (nat_trans_ax g). apply idpath. Qed.
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.