Proposition is_natural_preserves_hor_comp' {C₁ C₂ : category} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {Cm₁ : hor_comp D₁} {Cm₂ : hor_comp D₂} {F : C₁ ⟶ C₂} {FF : twosided_disp_functor F F D₁ D₂} (FFc : preserves_hor_comp Cm₁ Cm₂ FF) {x₁ x₂ y₁ y₂ z₁ z₂ : C₁} {vx : x₁ --> x₂} {vy : y₁ --> y₂} {vz : z₁ --> z₂} {h₁ : D₁ x₁ y₁} {k₁ : D₁ y₁ z₁} {h₂ : D₁ x₂ y₂} {k₂ : D₁ y₂ z₂} (s₁ : h₁ -->[ vx ][ vy ] h₂) (s₂ : k₁ -->[ vy ][ vz ] k₂) : transportb_disp_mor2 (id_left _ @ !(id_right _)) (id_left _ @ !(id_right _)) (#2 FF (double_hor_comp_mor Cm₁ s₁ s₂) ;;2 idtoiso_twosided_disp (idpath _) (idpath _) (FFc x₂ y₂ z₂ h₂ k₂)) = idtoiso_twosided_disp (idpath _) (idpath _) (FFc x₁ y₁ z₁ h₁ k₁) ;;2 double_hor_comp_mor Cm₂ (#2 FF s₁) (#2 FF s₂). Proof. rewrite is_natural_preserves_hor_comp. rewrite transportbf_disp_mor2. 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.