Definition disp_psfunctor_invertible_2cell : disp_invertible_2cell (_,, psfunctor_is_iso F α) (disp_psfunctor_mor _ _ _ FF ff) (disp_psfunctor_mor _ _ _ FF gg). Proof. repeat use tpair; cbn. - exact (disp_psfunctor_cell _ _ _ FF αα). - exact (disp_psfunctor_cell _ _ _ FF (disp_inv_cell αα)). - abstract (rewrite <- (disp_psfunctor_vcomp2_alt _ _ _ _ (pr2 FF)); rewrite disp_vcomp_rinv; rewrite disp_psfunctor_cell_transportb; unfold transportb; rewrite transport_f_f; rewrite (disp_psfunctor_id2 _ _ _ _ (pr2 FF)); unfold transportb; rewrite transport_f_f; apply (@transportf_paths _ (λ p, _ ==>[ p ] _)); apply B₂). - abstract (rewrite <- (disp_psfunctor_vcomp2_alt _ _ _ _ (pr2 FF)); rewrite disp_vcomp_linv; rewrite disp_psfunctor_cell_transportb; unfold transportb; rewrite transport_f_f; rewrite (disp_psfunctor_id2 _ _ _ _ (pr2 FF)); unfold transportb; rewrite transport_f_f; apply (@transportf_paths _ (λ p, _ ==>[ p ] _)); apply B₂). 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.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.FiberCategory. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Import PseudoFunctor.Notations.