Definition acti_to_acat_data : psfunctor_data ACTI ACAT. Proof. use make_psfunctor_data. - exact acti_to_acat_on_ob. - exact (λ a1 a2 f, acti_to_acat_on_mor f). - intros a1 a2 f g α. exists (pr1 α). intros v a. exact (eqtohomot (base_paths _ _ (pr2 α v)) a). - intro a. use tpair. + exists (λ c, identity c). intro ; intros ; exact (id_right _ @ ! id_left _). + intros v c. cbn. rewrite ! id_right. unfold identity_lineator_data. cbn. apply (! functor_id _ _). - intros a1 a2 a3 f g. use tpair. + exists (λ c, identity _). intro ; intros ; exact (id_right _ @ ! id_left _). + intros v c. cbn. unfold comp_lineator_data. cbn. rewrite ! id_right. rewrite ! id_left. cbn. rewrite (functor_id ((pr12 a3) v)). rewrite id_left. apply idpath. 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.