proposition: Lemma actfromprecomp_coprod_distributor_law : actegory_coprod_distributor_iso_law _ _ _ actfromprecomp_coprod_distributor_data.
proof: Proof. intros F Gs. split. - apply nat_trans_eq; [apply D |]. intro c. cbn. rewrite id_left. apply pathsinv0, Coproduct_endo_is_identity. intro i. unfold coproduct_nat_trans_data. cbn in Gs. apply (CoproductInCommutes I D (λ i0 : I, Gs i0 (pr1 F c)) (CP _) _ (λ i0 : I, coproduct_nat_trans_in_data I C D CP Gs i0 (pr1 F c)) i). - etrans. { apply postcompWithCoproductArrow. } etrans. 2: { apply pathsinv0, CoproductArrowEta. } apply maponpaths, funextsec; intro i; (rewrite id_right; apply nat_trans_eq; [apply D |]; intro c; apply id_right). Qed.
imports: UniMath.Foundations.PartD UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.Bicategories.MonoidalCategories.WhiskeredMonoidalFromBicategory UniMath.CategoryTheory.Actegories.Actegories UniMath.CategoryTheory.Actegories.ConstructionOfActegories UniMath.CategoryTheory.Actegories.MorphismsOfActegories UniMath.CategoryTheory.Monoidal.Examples.EndofunctorsMonoidalElementary UniMath.CategoryTheory.Actegories.Examples.ActionOfEndomorphismsInCATElementary UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Core.Examples.BicatOfCats UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Actegories.CoproductsInActegories