Proposition is_natural_preserves_hor_id {C₁ C₂ : category} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {I₁ : hor_id D₁} {I₂ : hor_id D₂} {F : C₁ ⟶ C₂} {FF : twosided_disp_functor F F D₁ D₂} (FFI : preserves_hor_id I₁ I₂ FF) {x y : C₁} (f : x --> y) : #2 FF (double_id_mor I₁ f) ;;2 idtoiso_twosided_disp (idpath _) (idpath _) (FFI y) = transportf_disp_mor2 (id_left _ @ !(id_right _)) (id_left _ @ !(id_right _)) (idtoiso_twosided_disp (idpath _) (idpath _) (FFI x) ;;2 double_id_mor I₂ (#F f)). Proof. exact (pr2 FFI x y f). 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.