proposition: Proposition transportf_hor_id_data {C : category} {D : twosided_disp_cat C C} {x y : C} {f : x --> y} {g : x --> y} {ψ₁ ψ₂ : ∏ (x : C), D x x} (p : ψ₁ = ψ₂) (fg : ψ₁ x -->[ f ][ g ] ψ₁ y) : transportf (λ (φ : ∏ (x : C), D x x), φ x -->[ f ][ g ] φ y) p fg = transportf_disp_mor2 (id_right _ @ id_left _) (id_right _ @ id_left _) (idtoiso_twosided_disp (idpath _) (idpath _) (!(toforallpaths _ _ _ p x)) ;;2 fg ;;2 idtoiso_twosided_disp (idpath _) (idpath _) (toforallpaths _ _ _ p y)).
proof: Proof. induction p ; cbn. rewrite (id_two_disp_right (D := D)). unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite (id_two_disp_left (D := D)). unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. refine (!_). apply transportf_disp_mor2_idpath. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.Categories.CategoryOfSetCategories UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Examples.StrictTwoSidedDispCat 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 UniMath.Bicategories.DoubleCategories.Basics.StrictDoubleCatBasics