Definition horizontal_dual_vertically_saturated (H : vertically_saturated B) : vertically_saturated horizontal_dual_verity_double_bicat. Proof. refine (λ (x y : B) (v₁ v₂ : x -|-> y), _). use isweq_iso. - exact (λ s, square_to_vertical_cell H s). - abstract (intro τ ; cbn ; refine (_ @ vertical_cell_to_square_to_vertical_cell H τ) ; apply maponpaths ; unfold vertical_cell_to_square ; rewrite lwhisker_id_h_square ; apply idpath). - abstract (intro s ; cbn ; refine (_ @ square_to_vertical_cell_to_square H s) ; unfold vertical_cell_to_square ; rewrite lwhisker_id_h_square ; apply idpath). 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.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. 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.