Lemma reindexed_lax_lineator_laws : lineator_laxlaws Mon_W (reindexed_actegory Mon_V ActC Mon_W U) (reindexed_actegory Mon_V ActD Mon_W U) H reindexed_lax_lineator_data. Proof. split4. - intro; intros. apply (lineator_linnatleft _ _ _ _ ll). - intro; intros. apply (lineator_linnatright _ _ _ _ ll). - intro; intros. cbn. unfold reindexed_lax_lineator_data, reindexed_actor_data. etrans. 2: { repeat rewrite assoc'. apply maponpaths. rewrite assoc. apply (lineator_preservesactor _ _ _ _ ll). } etrans. 2: { rewrite assoc. apply cancel_postcomposition. apply pathsinv0, lineator_linnatright. } etrans. 2: { rewrite assoc'. apply maponpaths. apply functor_comp. } apply idpath. - intro; intros. cbn. unfold reindexed_lax_lineator_data, reindexed_action_unitor_data. etrans. 2: { apply maponpaths. apply (lineator_preservesunitor _ _ _ _ ll). } etrans. 2: { rewrite assoc. apply cancel_postcomposition. apply pathsinv0, lineator_linnatright. } etrans. 2: { rewrite assoc'. apply maponpaths. apply functor_comp. } apply idpath. 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.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.whiskering. 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.ConstructionOfActegories. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Actegories.CoproductsInActegories. Require Import UniMath.CategoryTheory.Actegories.ProductsInActegories. Require Import UniMath.CategoryTheory.Actegories.ProductActegory. Import BifunctorNotations. Import MonoidalNotations. Import ActegoryNotations. Section ActegoryMorphismFromRelativeLaxCommutator. End ActegoryMorphismFromRelativeLaxCommutator.