Definition global_mor_comp : disp_invertible_2cell (id2_invertible_2cell _) (global_lift_mor (local_iso_cleaving_1cell local_D (ff ;; gg) (idempunitor y))) (local_iso_cleaving_1cell local_D (global_lift_mor ff ;; global_lift_mor gg) (idempunitor x)). Proof. simple refine (_ ,, _). - exact (cartesian_1cell_lift_2cell D _ (is_cartesian_mor_from_global_lift f yy₃) global_mor_comp_cell _ (local_iso_cleaving_1cell local_D (global_lift_mor ff ;; global_lift_mor gg) (idempunitor x) ,, global_mor_comp_factor)). - use (cartesian_1cell_lift_2cell_invertible D (is_cartesian_mor_from_global_lift f yy₃) _ _ _ (_ ,, _)). exact (pr2 global_mor_comp_cell). 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.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Discreteness. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.FiberCategory. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.Cartesians.