Definition univalent_2_0_weakly_hor_invariant (B : verity_double_bicat) (H : is_univalent_2_0 (hor_bicat_of_verity_double_bicat B)) : weakly_hor_invariant B. Proof. assert (∏ (x y : B) (h : adjoint_equivalence x y), ∑ (v : x -|-> y), are_companions h v) as H'. { use (J_2_0 H) ; cbn. intros x. exact (_ ,, id_are_companions x). } intros x y h Hh. exact (H' x y (h ,, Hh)). Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CellsAndSquares. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.UnderlyingCats. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.LocalUnivalence. Require Import UniMath.Bicategories.DoubleCategories.Examples.TransposeDoubleBicat.