Lemma bincoprod_distributor_commutes_with_associativity_of_coproduct (δ : actegory_bincoprod_distributor) (v : V) (c d e : C) : v ⊗^{Act}_{l} bincoprod_associator_data BCP c d e · δ v c (BCP d e) · #(bincoproduct_functor BCP) (catbinprodmor (identity (v ⊗_{Act} c)) (δ v d e)) = δ v (BCP c d) e · #(bincoproduct_functor BCP) (catbinprodmor (δ v c d) (identity (v ⊗_{Act} e))) · bincoprod_associator_data BCP (v ⊗_{Act} c) (v ⊗_{Act} d) (v ⊗_{Act} e). Proof. repeat rewrite assoc'. apply (z_iso_inv_to_left _ _ _ (_,,bincoprod_functor_lineator_strongly δ v (((BCP c d): C),, e))). repeat rewrite assoc. apply (z_iso_inv_to_right _ _ _ _ (functor_on_z_iso (functor_fix_fst_arg _ _ _ (bincoproduct_functor BCP) (v ⊗_{ Act} c)) (_,,bincoprod_functor_lineator_strongly δ v (d,,e)))). apply (z_iso_inv_to_right _ _ _ _ (_,,bincoprod_functor_lineator_strongly δ v (c,,((BCP d e): C)))). repeat rewrite assoc'. apply (z_iso_inv_to_left _ _ _ (functor_on_z_iso (functor_fix_snd_arg _ _ _ (bincoproduct_functor BCP) (v ⊗_{ Act} e)) (_,,bincoprod_functor_lineator_strongly δ v (c,,d)))). repeat rewrite assoc. apply actegory_bincoprod_antidistributor_commutes_with_associativity_of_coproduct. 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.