Definition disp_invertible_2cell_along_iso_weq : disp_invertible_2cell (id2_invertible_2cell (# F f)) (disp_psfunctor_mor _ _ _ FF ff) (disp_psfunctor_mor _ _ _ FF gg) ≃ disp_invertible_2cell (id2_invertible_2cell f) ff gg. Proof. use weq_iso. - exact disp_invertible_2cell_weq_along_iso_left. - exact disp_invertible_2cell_weq_along_iso_right. - abstract (intros τ ; use subtypePath ; [ intro ; apply isaprop_is_disp_invertible_2cell | ] ; cbn ; rewrite disp_psfunctor_iso_invmap_2_eq_1 ; rewrite transport_b_b ; unfold transportb ; refine (_ @ idpath_transportf (λ z, _ ==>[ z ] _) _) ; apply maponpaths_2 ; apply cellset_property). - abstract (intros τ ; use subtypePath ; [ intro ; apply isaprop_is_disp_invertible_2cell | ] ; cbn ; rewrite transport_b_b ; refine (_ @ disp_psfunctor_iso_invmap_2_eq_2 HF τ) ; apply maponpaths ; unfold transportb ; 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.Core.Functors. 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.Examples.Initial. Require Import UniMath.Bicategories.Core.Examples.Final. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.TransportLaws. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor. Require Import UniMath.Bicategories.DisplayedBicats.DispTransportLaws.