Lemma coprod_functor_lineator_laxlaws (δ : actegory_coprod_distributor) : lineator_laxlaws _ _ _ _ (coprod_functor_lineator_data δ). Proof. red; repeat split. - red. intros v cs1 cs2 g. apply (z_iso_inv_to_right _ _ _ _ (_,,coprod_functor_lineator_strongly δ v _)). rewrite assoc'. apply (z_iso_inv_to_left _ _ _ (_,,coprod_functor_lineator_strongly δ v _)). apply actegory_coprod_antidistributor_nat_left. - red. intros v1 v2 cs f. apply (z_iso_inv_to_right _ _ _ _ (_,,coprod_functor_lineator_strongly δ _ _)). rewrite assoc'. apply (z_iso_inv_to_left _ _ _ (_,,coprod_functor_lineator_strongly δ _ _)). apply actegory_coprod_antidistributor_nat_right. - red. intros v w cs. apply pathsinv0, (z_iso_inv_to_right _ _ _ _ (_,,coprod_functor_lineator_strongly δ _ _)). rewrite assoc'. apply (z_iso_inv_to_left _ _ _ (_,,coprod_functor_lineator_strongly δ _ _)). rewrite assoc. apply (z_iso_inv_to_right _ _ _ _ (functor_on_z_iso (leftwhiskering_functor Act v) (_,,coprod_functor_lineator_strongly δ _ _))). apply pathsinv0, coprod_antidistributor_pentagon_identity. - red. intro cs. apply pathsinv0, (z_iso_inv_to_left _ _ _ (_,,coprod_functor_lineator_strongly δ _ _)). apply pathsinv0, coprod_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.