proposition: 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: Proof. exact (pr2 FFI x y f). Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.Categories.CategoryOfSetCategories UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.TransportLaws UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics