proposition: Lemma is_functor_coproduct_of_functors_data : is_functor coproduct_of_functors_data.
proof: Proof. split; simpl; intros. - unfold functor_idax; intros; simpl in *. apply pathsinv0. apply Coproduct_endo_is_identity; intro i. unfold coproduct_of_functors_mor. eapply pathscomp0; [apply (CoproductOfArrowsIn _ _ (HD (λ i, (F i) a)))|]. now simpl; rewrite functor_id, id_left. - unfold functor_compax; simpl; unfold coproduct_of_functors_mor. intros; simpl in *. apply pathsinv0. eapply pathscomp0. apply CoproductOfArrows_comp. apply maponpaths, funextsec; intro i. now rewrite functor_comp. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.ProductCategory UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.whiskering