Proposition square_to_horizontal_cell_id (HB : horizontally_saturated B) {x y : B} (h : x --> y) : square_to_horizontal_cell HB (id_v_square_bicat h) = id2 h. Proof. use (invmaponpathsweq (horizontal_cell_to_square_weq HB h h)) ; cbn. rewrite square_to_horizontal_cell_to_square. rewrite horizontal_cell_to_square_id. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Discrete. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws.