Definition all_companions_adjequiv {B : verity_double_bicat} (H : vertically_saturated B) (H' : all_companions B) {x y : B} {h : x --> y} {v : x -|-> y} (c : are_companions h v) (Hh : left_adjoint_equivalence h) : left_adjoint_equivalence v. Proof. use all_equivs_companions_adjequiv. - exact H. - use all_companions_to_weakly_hor_invariant. exact H'. - exact h. - exact c. - exact 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.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. 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.CompanionPairs.