Proposition isaprop_companion_pair {B : verity_double_bicat} (H : vertically_saturated B) (HB_2_1 : locally_univalent_verity_double_bicat B) {x y : B} (h : x --> y) : isaprop (∑ (v : x -|-> y), are_companions h v). Proof. use invproofirrelevance. intros φ₁ φ₂. use (eq_companion_of_hor H HB_2_1). - use (v_sq_isotoid_2_1 H HB_2_1). use make_invertible_vertical_square. + use make_invertible_vertical_square_data. * exact (square_between_companions (pr2 φ₁) (pr2 φ₂)). * exact (square_between_companions (pr2 φ₂) (pr2 φ₁)). + split. * apply comp_square_between_companions. * apply comp_square_between_companions. - rewrite v_sq_idtoiso_isotoid_2_1 ; cbn. apply square_between_companions_unit. - rewrite v_sq_idtoiso_isotoid_2_1 ; cbn. apply square_between_companions_counit. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. 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.DoubleBicat.CompanionPairs.