Definition disp_psfunctor_iso_id_mor_weq {x : B₁} (xx yy : D₁ x) : xx -->[ id₁ x ] yy ≃ FF x xx -->[ id₁ (F x) ] FF x yy. Proof. refine (_ ∘ disp_psfunctor_iso_mor_weq (id₁ x) xx yy)%weq. use weq_iso. - refine (λ f, transportb (λ z, _ -->[ z ] _) _ f). exact (isotoid_2_1 HB₂ (psfunctor_id F x)). - refine (λ f, transportf (λ z, _ -->[ z ] _) _ f). exact (isotoid_2_1 HB₂ (psfunctor_id F x)). - abstract (intro f ; cbn ; rewrite transportfbinv ; apply idpath). - abstract (intro f ; cbn ; rewrite transportbfinv ; apply idpath). 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.