Definition global_mor_comp_factor : disp_invertible_2cell (id2_invertible_2cell (id₁ x · f)) (local_iso_cleaving_1cell local_D (global_lift_mor ff ;; global_lift_mor gg) (idempunitor x) ;; mor_from_global_lift f yy₃) global_mor_comp_mor. Proof. refine (transportf (λ z, disp_invertible_2cell z _ _) _ (vcomp_disp_invertible (vcomp_disp_invertible (disp_invertible_2cell_rwhisker _ (disp_local_iso_cleaving_invertible_2cell local_D _ _)) (vcomp_disp_invertible (disp_invertible_2cell_rassociator _ _ _) (vcomp_disp_invertible (disp_invertible_2cell_lwhisker _ (pr2 (cartesian_1cell_lift_1cell D (mor_from_global_lift f yy₃) _ _))) (vcomp_disp_invertible (disp_invertible_2cell_lwhisker _ (disp_local_iso_cleaving_invertible_2cell local_D _ _)) (vcomp_disp_invertible (disp_invertible_2cell_lassociator _ _ _) (disp_invertible_2cell_rwhisker _ (vcomp_disp_invertible (pr2 (cartesian_1cell_lift_1cell D _ _ _)) (disp_local_iso_cleaving_invertible_2cell local_D _ _)))))))) (inverse_of_disp_invertible_2cell (disp_local_iso_cleaving_invertible_2cell local_D _ _)))). abstract (use subtypePath ; [ intro ; apply isaprop_is_invertible_2cell | ] ; cbn ; rewrite lwhisker_id2, !id2_left ; rewrite <- !lwhisker_vcomp, <- !rwhisker_vcomp ; rewrite !vassocl ; rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ; rewrite lunitor_lwhisker ; rewrite runitor_lunitor_identity ; rewrite !vassocr ; rewrite rwhisker_vcomp ; rewrite linvunitor_lunitor ; rewrite id2_rwhisker ; rewrite id2_left ; rewrite !vassocl ; rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ; rewrite lunitor_triangle ; rewrite !vassocr ; rewrite vcomp_lunitor ; use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ; rewrite id2_left ; refine (_ @ id2_right _) ; rewrite !vassocl ; apply maponpaths ; use vcomp_move_R_pM ; [ is_iso | ] ; cbn ; rewrite id2_right ; refine (_ @ id2_left _) ; rewrite !vassocr ; apply maponpaths_2 ; rewrite <- runitor_triangle ; rewrite runitor_lunitor_identity ; rewrite lunitor_lwhisker ; rewrite rwhisker_vcomp ; rewrite rinvunitor_runitor ; apply id2_rwhisker). 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.