Definition actions_disp_locally_groupoid : disp_locally_groupoid actions_disp_bicat. Proof. red. intros A A' F F' invertibleη actn actn' ζ ζ' Hypη. use tpair. - intros a v. red. assert (Hypηinst := Hypη a v). red in Hypηinst. apply pathsinv0. set (η_nat_z_iso := nat_z_iso_from_z_iso (homset_property A') invertibleη). set (η_nat_z_iso_inst1 := nat_z_iso_pointwise_z_iso η_nat_z_iso (ActionBasedStrongFunctorCategory.odot Mon_V actn (a, v))). apply (z_iso_inv_on_left _ _ _ _ η_nat_z_iso_inst1). rewrite <- assoc. set (η_nat_z_iso_inst2 := nat_z_iso_pointwise_z_iso η_nat_z_iso a). set (aux1_z_iso := precatbinprod_z_iso η_nat_z_iso_inst2 (identity_z_iso v)). set (aux2_z_iso := functor_on_z_iso (ActionBasedStrongFunctorCategory.odot' Mon_V actn') aux1_z_iso). apply pathsinv0. apply (z_iso_inv_on_right _ _ _ aux2_z_iso). exact Hypηinst. - split; apply actions_disp_2cells_isaprop. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorsTensored. Require Import UniMath.Bicategories.MonoidalCategories.Actions. Require Import UniMath.Bicategories.MonoidalCategories.ActionBasedStrength. Require Import UniMath.Bicategories.MonoidalCategories.ActionBasedStrongFunctorCategory. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Import Bicat.Notations.