Proposition lift_transportf {x y : E} {f g : x --> y} {τ₁ τ₂ : f ==> g} (p : τ₁ = τ₂) {xx : lift_disp_prebicat_data x} {yy : lift_disp_prebicat_data y} {ff : xx -->[ f ] yy} {gg : xx -->[ g ] yy} (ττ : ff ==>[ τ₁ ] gg) : transportf (λ (τ : f ==> g), ff ==>[ τ ] gg) p ττ = transportf (λ (τ : pr1 f ==> pr1 g), ff ==>[ τ ] gg) (maponpaths pr1 p) ττ. Proof. induction p. apply idpath. Qed.
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.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Section FromLift. End FromLift.