Definition global_mor_id_cell : disp_invertible_2cell (rwhisker_of_invertible_2cell _ (id2_invertible_2cell _)) (local_iso_cleaving_1cell local_D (mor_from_global_lift f yy ;; id_disp yy) (comp_of_invertible_2cell (lunitor_invertible_2cell f) (rinvunitor_invertible_2cell f))) global_mor_id_mor. Proof. refine (transportf (λ z, disp_invertible_2cell z _ _) _ (vcomp_disp_invertible (disp_local_iso_cleaving_invertible_2cell local_D (mor_from_global_lift f yy ;; id_disp yy) (comp_of_invertible_2cell (lunitor_invertible_2cell _) (rinvunitor_invertible_2cell _))) (vcomp_disp_invertible (disp_invertible_2cell_runitor _) (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 ; rewrite !vassocl ; rewrite (maponpaths (λ z, _ • z) (vassocr _ _ _)) ; rewrite rinvunitor_runitor ; rewrite id2_left ; rewrite id2_rwhisker ; 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.