proposition: Lemma is_nat_trans_coproduct_nat_trans_data : is_nat_trans _ _ coproduct_nat_trans_data.
proof: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.ProductCategory