Proposition disp_psfunctor_iso_invmap_2_vcomp {x y : B₁} {f : x --> y} {xx : D₁ x} {yy : D₁ y} {ff gg hh : xx -->[ f ] yy} (ττ : disp_psfunctor_mor D₁ D₂ F FF ff ==>[ ## F (id2 f)] disp_psfunctor_mor D₁ D₂ F FF gg) (θθ : disp_psfunctor_mor D₁ D₂ F FF gg ==>[ ## F (id2 f)] disp_psfunctor_mor D₁ D₂ F FF hh) : disp_psfunctor_iso_invmap_2 ττ •• disp_psfunctor_iso_invmap_2 θθ = transportb (λ z, _ ==>[ z ] _) (id2_left _) (disp_psfunctor_iso_invmap_2 (transportb (λ z, _ ==>[ z ] _) (maponpaths (λ h, ##F h) (!(id2_left _)) @ psfunctor_vcomp F _ _) (ττ •• θθ))). Proof. simple refine (!(transportfbinv (λ z, _ ==>[ z ] _) (!(id2_left _)) (disp_psfunctor_iso_invmap_2 ττ •• disp_psfunctor_iso_invmap_2 θθ)) @ _). unfold transportb. apply maponpaths. etrans. { exact (!(disp_psfunctor_iso_invmap_2_eq_2 _)). } apply maponpaths. etrans. { apply disp_psfunctor_cell_transportf. } etrans. { apply maponpaths. apply (disp_psfunctor_vcomp2 _ _ _ FF (pr2 FF)). } unfold transportb. rewrite transport_f_f. rewrite !disp_psfunctor_iso_invmap_2_eq_1. apply maponpaths_2. apply cellset_property. Qed.
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.