Definition fmonoidal_stronglaws_to_object (Act : actegory Mon_V C) : fmonoidal_stronglaws (pr1 (fmonoidal_data_to_object Act)) (pr2 (fmonoidal_data_to_object Act)). Proof. repeat (use tpair). - intros v w. use tpair. + exists (λ c, actegory_actordata Act w v c). abstract (intros c1 c2 f; cbn; apply (! actegory_actornatleft _ _ _ _ _ _ _ )). + use tpair; [ apply nat_trans_eq; [ apply homset_property | intro; apply actegory_actorisolaw] | apply nat_trans_eq; [ apply homset_property | intro; apply (pr1 (actegory_actorisolaw Mon_V Act w v x))] ]. - exact (λ c, actegory_unitordata Act c). - abstract (intros c1 c2 f; cbn; apply actegory_unitornat). - abstract (use nat_trans_eq; [ apply homset_property | intro; cbn; apply actegory_unitorisolaw]). - abstract (use nat_trans_eq; [ apply homset_property | intro; cbn; apply actegory_unitorisolaw]). Defined.
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.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. 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.Bicategories.MonoidalCategories.BicatOfActegories. Require Import UniMath.Bicategories.MonoidalCategories.BicatOfActionsInBicat. Require Import UniMath.Bicategories.MonoidalCategories.WhiskeredMonoidalFromBicategory. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Import BifunctorNotations. Section ActegoryFromObject. End ActegoryFromObject. Section FromActegoriesToActionsInCat. Import MonoidalNotations. End FromActegoriesToActionsInCat. Section FromActionInCatToActegories. End FromActionInCatToActegories.