Proposition square_to_horizontal_cell_comp (HB : horizontally_saturated B) {x y : B} {h₁ h₂ h₃ : x --> y} (s₁ : square_double_bicat h₁ h₂ (id_v x) (id_v y)) (s₂ : square_double_bicat h₂ h₃ (id_v x) (id_v y)) : square_to_horizontal_cell HB (comp_hor_globular_square s₁ s₂) = square_to_horizontal_cell HB s₁ • square_to_horizontal_cell HB s₂. Proof. use (invmaponpathsweq (horizontal_cell_to_square_weq HB h₁ h₃)) ; cbn. rewrite horizontal_cell_to_square_comp. rewrite !square_to_horizontal_cell_to_square. 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.