Definition double_cat_to_verity_double_bicat_are_companions {C : double_cat} {x y : C} (h : x -->h y) (v : x -->v y) : double_cat_are_companions h v ≃ are_companions (B := double_cat_to_verity_double_bicat C) v h. Proof. use weqfibtototal. intro φ. use weqfibtototal. intro ψ. use weqimplimpl. - abstract (intros p ; induction p as [ p₁ p₂ ] ; refine (p₁ ,, _) ; refine (_ @ p₂) ; cbn -[transportf_square] ; do 2 use transportf_square_eq ; apply idpath). - abstract (cbn -[transportf_square] ; intros p ; induction p as [ p₁ p₂ ] ; refine (p₁ ,, _) ; refine (_ @ p₂) ; do 2 use transportf_square_eq ; apply idpath). - abstract (apply isapropdirprod ; apply isaset_square). - abstract (apply isapropdirprod ; apply isaset_square). 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.Examples.DiscreteBicat. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.Core. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws.TransportLaws. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints. Require Import UniMath.Bicategories.DoubleCategories.Underlying.HorizontalBicategory.