Proposition eq_are_companions {B : verity_double_bicat} {x y : B} {h : x --> y} {v : x -|-> y} {c₁ c₂ : are_companions h v} (p : unit_are_companions c₁ = unit_are_companions c₂) (q : counit_are_companions c₁ = counit_are_companions c₂) : c₁ = c₂. Proof. use total2_paths_f. - exact p. - use subtypePath. { intro. apply isapropdirprod ; apply isaset_square_double_bicat. } rewrite pr1_transportf. rewrite transportf_const. exact q. Qed.
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.