Definition locally_groupoid_grpds : locally_groupoid grpds. Proof. intros G₁ G₂ F₁ F₂ α. use make_is_invertible_2cell. - refine (_ ,, tt). use make_nat_trans. + exact (λ x, inv_from_z_iso (_ ,, pr2 G₂ _ _ (pr11 α x))). + abstract (intros x y f ; cbn ; refine (!_) ; apply z_iso_inv_on_right ; rewrite !assoc ; apply z_iso_inv_on_left ; simpl ; exact (!(pr21 α x y f))). - abstract (apply subtypePath ; try (intro ; apply isapropunit) ; apply nat_trans_eq ; try apply homset_property ; intro x ; cbn ; exact (z_iso_inv_after_z_iso (_ ,, _))). - abstract (apply subtypePath ; try (intro ; apply isapropunit) ; apply nat_trans_eq ; try apply homset_property ; intro x ; cbn ; exact (z_iso_after_z_iso_inv (_ ,, _))). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub.