Definition acti_to_acat_counit_data : pstrans_data (Composition.comp_psfunctor acti_to_acat acat_to_acti) (Identity.id_psfunctor ACAT). Proof. use make_pstrans_data. - exact (λ a, acti_to_acat_counit_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 ). * abstract ( intro ; intro ; simpl ; unfold comp_lineator_data ; simpl ; rewrite functor_id ; rewrite ! id_right ; rewrite (bifunctor_leftid (_ ,, pr122 a2)); rewrite ! id_left ; apply idpath ). + use tpair. * use tpair. -- exists (λ c, identity _). abstract ( intros c1 c2 g ; simpl ; rewrite id_left ; apply id_right ). -- abstract ( intro ; intro ; simpl ; unfold comp_lineator_data ; simpl ; rewrite functor_id ; rewrite ! id_right ; rewrite (bifunctor_leftid (_ ,, pr122 a2)) ; rewrite ! id_left ; apply idpath ). * use tpair ; use equality_of_2cells_in_ACAT ; 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.