Definition ACAT_invertible_2cell {a1 a2 : ACAT} (f : ACAT⟦a1,a2⟧) : invertible_2cell (ps_base_ACAT a1 · acti_to_acat_on_mor (acat_to_acti_on_mor f)) (f · ps_base_ACAT a2). Proof. repeat (use tpair). - exact (λ _, identity _). - abstract ( intro ; intros ; cbn ; rewrite id_left ; apply id_right ). - abstract ( intro ; intro ; cbn ; unfold comp_lineator_data ; simpl ; rewrite (bifunctor_leftid (_ ,, pr122 a2)) ; rewrite ! id_left ; rewrite functor_id ; rewrite id_right ; apply id_right ). - exact (λ _, identity _). - abstract ( intro ; intros ; cbn ; rewrite id_left ; apply id_right ). - abstract ( intro ; intro ; cbn ; unfold comp_lineator_data ; simpl ; rewrite (bifunctor_leftid (_ ,, pr122 a2)) ; rewrite ! id_left ; rewrite functor_id ; apply idpath ). - abstract (use equality_of_2cells_in_ACAT; intro; cbn; apply id_right). - abstract (use equality_of_2cells_in_ACAT; intro; cbn; 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.