Definition acti_to_acat_unit_data : pstrans_data (Composition.comp_psfunctor acat_to_acti acti_to_acat) (Identity.id_psfunctor ACTI). Proof. use make_pstrans_data. - exact (λ a, acti_to_acat_unit_data_on_ob a). - intros a1 a2 f. use tpair. + use tpair. * exists (λ c, identity _). intros c1 c2 g. abstract ( simpl ; rewrite id_left ; apply id_right ). * intro. use nat_trans_eq. { apply homset_property. } intro. abstract ( cbn ; rewrite ! id_left ; rewrite ! id_right ; unfold ActionBasedStrongFunctorsWhiskeredMonoidal.parameterized_distributivity_bicat_nat_funclass ; rewrite (functor_id (pr1 f)) ; rewrite (functor_id ((pr12 a2) v)) ; rewrite id_left ; apply id_right ). + use tpair. * use tpair. -- exists (λ c, identity _). intros c1 c2 g. simpl. rewrite id_left. apply id_right. -- intro. use nat_trans_eq. { apply homset_property. } intro. abstract ( simpl ; rewrite ! id_left ; rewrite ! id_right ; rewrite (functor_id ((pr12 a2) v)) ; rewrite (functor_id (pr1 f)) ; rewrite id_left ; apply (! id_right _) ). * use tpair ; use equality_of_2cells_in_ACTI ; intro ; simpl ; apply id_right. 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.