Definition acti_to_acat_counit_data_on_ob (a : ACAT) : ACAT ⟦Composition.comp_psfunctor acti_to_acat acat_to_acti a, Identity.id_psfunctor ACAT a⟧. Proof. exists (identity _). exists (λ _ _, identity _). abstract ( simpl ; repeat split ; (intro ; intros ; try (rewrite id_right ; rewrite id_left ; apply idpath)) ; [ rewrite id_left ; rewrite id_right ; rewrite (bifunctor_leftid (_ ,, pr122 a)) ; apply (! id_right _) | rewrite id_left ; apply idpath ]). 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.