Definition double_cat_vertically_saturated : vertically_saturated double_cat_to_verity_double_bicat. Proof. refine (λ (x y : C) (h₁ h₂ : x -->h y), _). use isweq_iso. - exact (idfun _). - abstract (intros s ; cbn ; refine (maponpaths (transportf_square _ _) (square_id_right_v _) @ _) ; unfold transportb_square ; rewrite transportf_f_square ; apply transportf_square_id). - abstract (intros s ; cbn ; refine (maponpaths (transportf_square _ _) (square_id_right_v _) @ _) ; unfold transportb_square ; rewrite transportf_f_square ; apply transportf_square_id). 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.