Definition pb_ump_1_grpds_iso_comma : pb_ump_1 (grpds_iso_comma_pb_cone F G). Proof. intro q. use make_pb_1cell. - refine (_ ,, tt). use iso_comma_ump1. + exact (pr1 (pb_cone_pr1 q)). + exact (pr1 (pb_cone_pr2 q)). + exact (grpds_2cell_to_nat_z_iso (pr1 (pb_cone_cell q))). - use make_invertible_2cell. + refine (_ ,, tt). apply iso_comma_ump1_pr1. + apply locally_groupoid_grpds. - use make_invertible_2cell. + refine (_ ,, tt). apply iso_comma_ump1_pr2. + apply locally_groupoid_grpds. - abstract (use subtypePath ; [ intro ; apply isapropunit | ] ; use nat_trans_eq ; [ apply homset_property | ] ; intros x ; cbn ; unfold pb_cone_cell ; rewrite (functor_on_inv_from_z_iso (pr1 G)) ; rewrite (functor_id (pr1 F)) ; rewrite !id_left, id_right ; refine (!(id_right _) @ _) ; apply maponpaths ; use inv_z_iso_unique' ; unfold precomp_with ; cbn ; rewrite id_right ; apply functor_id). 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.IsoCommaCategory. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.Groupoids. Require Import UniMath.Bicategories.Limits.Pullbacks.