Definition eq_cartesian_lift {B : bicat} (D : disp_bicat B) (HD : disp_univalent_2_1 D) {x y : B} {yy : D y} {f : x --> y} (ℓ₁ ℓ₂ : ∑ (xx : D x) (ff : xx -->[ f ] yy), cartesian_1cell D ff) (p₁ : pr1 ℓ₁ = pr1 ℓ₂) (p₂ : transportf (λ z, z -->[ f ] yy) p₁ (pr12 ℓ₁) = pr12 ℓ₂) : ℓ₁ = ℓ₂. Proof. induction ℓ₁ as [ xx₁ [ ff₁ Hff₁ ]]. induction ℓ₂ as [ xx₂ [ ff₂ Hff₂ ]]. cbn in *. induction p₁ ; cbn in *. induction p₂ ; cbn in *. do 2 apply maponpaths. apply isaprop_cartesian_1cell. apply HD. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Opposite. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.Cartesians. Require Import UniMath.Bicategories.DisplayedBicats.EquivalenceBetweenCartesians.