Definition ACTI_invertible_2cell {a1 a2 : ACTI} (f : ACTI⟦a1,a2⟧) : invertible_2cell (ps_base_ACTI a1 · acat_to_acti_on_mor (acti_to_acat_on_mor f)) (f · ps_base_ACTI a2). Proof. use tpair. - use tpair. + exists (λ _, identity _). abstract ( intro ; intros ; cbn ; rewrite id_left ; apply id_right ). + intro. use nat_trans_eq. { apply homset_property. } abstract ( intro ; cbn ; rewrite (functor_id (pr1 f)) ; rewrite ! id_left ; rewrite ! id_right ; rewrite (functor_id ((pr12 a2) v)) ; apply (! id_left _) ). - use tpair. + use tpair. * exists (λ _, identity _). abstract ( intro ; intros ; rewrite id_left ; apply id_right ). * intro. use nat_trans_eq. { apply homset_property. } abstract ( intro ; cbn ; rewrite (functor_id (pr1 f)) ; rewrite ! id_left ; rewrite ! id_right ; rewrite (functor_id ((pr12 a2) v)) ; apply (! id_left _) ). + abstract ( use tpair ; use equality_of_2cells_in_ACTI ; intro ; simpl ; apply id_left). 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.