Lemma path_is_hor_gregarious_equivalence (H : vertically_saturated B) (HB_2_1 : locally_univalent_verity_double_bicat B) {x y : B} {h : x --> y} (φ₁ φ₂ : is_hor_gregarious_equivalence h) (p : (φ₁ : x -|-> y) = φ₂) (q₁ : lunitor _ ▿s (linvunitor _ ▵s v_sq_idtoiso_2_1 p ⋆h unit_are_companions φ₂) = unit_are_companions φ₁) (q₂ : runitor _ ▿s (linvunitor _ ▵s counit_are_companions φ₁ ⋆h v_sq_idtoiso_2_1 p) = counit_are_companions φ₂) : φ₁ = φ₂. Proof. use pathsdirprod. - apply isaprop_left_adjoint_equivalence. apply HB_2_1. - use subtypePath. { intro. apply isaprop_left_adjoint_equivalence. apply HB_2_1. } exact (eq_companion_of_hor H HB_2_1 p q₁ q₂). Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Discrete. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.UnderlyingCats. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CellsAndSquares. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.LocalUnivalence. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CompanionPairs. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CompanionPairUnique. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.CompanionPairAdjEquiv.