Proposition composition_preserves_hor_id_data {C₁ C₂ C₃ : category} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {D₃ : twosided_disp_cat C₃ C₃} {I₁ : hor_id D₁} {I₂ : hor_id D₂} {I₃ : hor_id D₃} {F : C₁ ⟶ C₂} {G : C₂ ⟶ C₃} {FF : twosided_disp_functor F F D₁ D₂} (FFI : preserves_hor_id I₁ I₂ FF) {GG : twosided_disp_functor G G D₂ D₃} (GGI : preserves_hor_id I₂ I₃ GG) : preserves_hor_id_data I₁ I₃ (comp_twosided_disp_functor FF GG). Proof. intro x ; cbn. refine (_ @ GGI (F x)). apply maponpaths. apply (FFI x). Defined.
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.