Proposition identity_preserves_hor_comp_laws {C : category} {D : twosided_disp_cat C C} (Cm : hor_comp D) : preserves_hor_comp_laws (identity_preserves_hor_comp_data Cm). Proof. intros x₁ x₂ y₁ y₂ z₁ z₂ vx vy vz h₁ k₁ h₂ k₂ s₁ s₂ ; cbn. rewrite id_two_disp_left, id_two_disp_right. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. use transportf_disp_mor2_eq. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TransportLaws. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics.