Lemma actegory_from_precomp_laws : actegory_laws Mon_endo actegory_from_precomp_data. Proof. split5. - exact action_from_precomp_laws. - split. + intros f g β. cbn. apply vcomp_lunitor. + split. * cbn. apply lunitor_linvunitor. * cbn. apply linvunitor_lunitor. - split4. + intros v w f f' β. cbn. apply lwhisker_lwhisker_rassociator. + intros v v' w f α. cbn. apply pathsinv0, rwhisker_rwhisker_alt. + intros v w w' f α. cbn. apply rwhisker_lwhisker_rassociator. + split. * cbn. apply rassociator_lassociator. * cbn. apply lassociator_rassociator. - intros v f. cbn. apply lunitor_lwhisker. - intros w v v' f. cbn. apply rassociator_rassociator. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.Bicategories.MonoidalCategories.WhiskeredMonoidalFromBicategory. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.Actegories.MorphismsOfActegories. Require Import UniMath.CategoryTheory.Monoidal.Examples.EndofunctorsMonoidalElementary. Require Import UniMath.CategoryTheory.Actegories.Examples.ActionOfEndomorphismsInCATElementary. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Actegories.CoproductsInActegories. Import Bicat.Notations. Import BifunctorNotations. Section Action_From_Precomposition. End Action_From_Precomposition.