Definition global_lift_mor {yy₁ yy₂ : D y} (ff : yy₁ -->[ id₁ y ] yy₂) : global_lift f yy₁ -->[ id₁ x ] global_lift f yy₂ := cartesian_1cell_lift_1cell D _ (is_cartesian_mor_from_global_lift f yy₂) (local_iso_cleaving_1cell local_D (mor_from_global_lift f yy₁ ;; ff) (comp_of_invertible_2cell (lunitor_invertible_2cell _) (rinvunitor_invertible_2cell _))). Section OnId. Context (yy : D y). Local Definition global_mor_id_mor : global_lift f yy -->[ id₁ x · f] yy := local_iso_cleaving_1cell local_D (mor_from_global_lift f yy) (lunitor_invertible_2cell _). Local Definition global_mor_id_factor : disp_invertible_2cell (id2_invertible_2cell (id₁ x · f)) (id_disp (global_lift f yy) ;; mor_from_global_lift f yy) global_mor_id_mor. Proof. refine (transportf (λ z, disp_invertible_2cell z _ _) _ (vcomp_disp_invertible (disp_invertible_2cell_lunitor _) (inverse_of_disp_invertible_2cell (disp_local_iso_cleaving_invertible_2cell local_D (mor_from_global_lift f yy) (lunitor_invertible_2cell f))))). abstract (use subtypePath ; [ intro ; apply isaprop_is_invertible_2cell | ] ; cbn ; apply lunitor_linvunitor). 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.