Definition counit_left_adjoint_data : Adjunctions.left_adjoint_data acti_to_acat_counit. Proof. exists ps_functor_ACAT. use tpair ; refine (_ ,, tt). - use tpair. + use tpair. * intro. use tpair. -- exists (λ _, identity _). abstract ( intro ; intros ; simpl ; rewrite id_left ; rewrite id_right ; apply idpath ). -- abstract ( intro ; intros ; cbn ; rewrite (bifunctor_leftid (_ ,, pr122 x)) ; rewrite ! id_left ; apply (! id_right _) ). * abstract ( intro ; intros ; use equality_of_2cells_in_ACAT ; intro ; cbn ; rewrite ! id_left ; rewrite id_right ; apply (! functor_id _ _) ). + exact (tt,,(tt,,tt)). - use tpair. + use tpair. * intro. use tpair. -- exists (λ _, identity _). abstract ( intro ; intros ; rewrite id_left ; apply id_right ). -- abstract ( intro ; intros ; simpl ; rewrite (bifunctor_leftid (_ ,, pr122 x)) ; apply id_right ). * abstract ( intro ; intros ; use equality_of_2cells_in_ACAT ; intro ; cbn ; rewrite ! id_left ; rewrite id_right ; apply (! functor_id _ _)). + exact (tt,,(tt,,tt)). 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.