Lemma bincoprod_functor_lineator_laxlaws (δ : actegory_bincoprod_distributor) : lineator_laxlaws _ _ _ _ (bincoprod_functor_lineator_data δ). Proof. red; repeat split. - red. intros v cd1 cd2 g. apply (z_iso_inv_to_right _ _ _ _ (_,,bincoprod_functor_lineator_strongly δ v _)). rewrite assoc'. apply (z_iso_inv_to_left _ _ _ (_,,bincoprod_functor_lineator_strongly δ v _)). apply (bincoprod_antidistributor_nat_left _ _ Act). - red. intros v1 v2 cd f. apply (z_iso_inv_to_right _ _ _ _ (_,,bincoprod_functor_lineator_strongly δ _ _)). rewrite assoc'. apply (z_iso_inv_to_left _ _ _ (_,,bincoprod_functor_lineator_strongly δ _ _)). apply (bincoprod_antidistributor_nat_right _ _ Act). - red. intros v w cd. apply pathsinv0, (z_iso_inv_to_right _ _ _ _ (_,,bincoprod_functor_lineator_strongly δ _ _)). rewrite assoc'. apply (z_iso_inv_to_left _ _ _ (_,,bincoprod_functor_lineator_strongly δ _ _)). rewrite assoc. apply (z_iso_inv_to_right _ _ _ _ (functor_on_z_iso (leftwhiskering_functor Act v) (_,,bincoprod_functor_lineator_strongly δ _ _))). apply pathsinv0, bincoprod_antidistributor_pentagon_identity. - red. intro cd. apply pathsinv0, (z_iso_inv_to_left _ _ _ (_,,bincoprod_functor_lineator_strongly δ _ _)). apply pathsinv0, bincoprod_antidistributor_triangle_identity. 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.