Lemma coprod_antidistributor_triangle_identity (cs : power_category I C) : #(coproduct_functor I CP) au^{actegory_power Mon_V I Act}_{cs} = actegory_coprod_antidistributor I_{Mon_V} cs · au^{Act}_{coproduct_functor I CP cs}. Proof. etrans. 2: { apply pathsinv0, postcompWithCoproductArrow. } cbn. unfold CoproductOfArrows. apply maponpaths, funextsec; intro i; apply pathsinv0, actegory_unitornat. 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.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.MorphismsOfActegories. Require Import UniMath.CategoryTheory.Actegories.ProductActegory. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.ProductCategory. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Import BifunctorNotations. Import MonoidalNotations. Import ActegoryNotations.