Definition strict_fiber_bicat_invertible_2cell {B : bicat} {D : disp_bicat B} {HD : local_iso_cleaving D} {b : B} {x y : strict_fiber_bicat D HD b} {f g : x --> y} {α : f ==> g} (Hα : is_disp_invertible_2cell (id2_invertible_2cell _) α) : is_invertible_2cell α. Proof. pose (αcell := (α ,, Hα) : disp_invertible_2cell (id2_invertible_2cell _) f g). use make_is_invertible_2cell. - exact (disp_inv_cell αcell). - abstract (cbn ; refine (maponpaths _ (disp_vcomp_rinv αcell) @ _) ; unfold transportb ; rewrite transport_f_f ; refine (_ @ idpath_transportf (λ z, _ ==>[ z ] _) _) ; apply maponpaths_2 ; apply cellset_property). - abstract (cbn ; refine (maponpaths _ (disp_vcomp_linv αcell) @ _) ; unfold transportb ; rewrite transport_f_f ; refine (_ @ idpath_transportf (λ z, _ ==>[ z ] _) _) ; apply maponpaths_2 ; apply cellset_property). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.FiberCategory. Require Import UniMath.Bicategories.DisplayedBicats.FiberBicategory.FiberBicategory1.