proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Actegories.Actegories UniMath.CategoryTheory.Actegories.MorphismsOfActegories UniMath.CategoryTheory.Actegories.ProductActegory UniMath.CategoryTheory.Actegories.ConstructionOfActegories UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.ProductCategory UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts