proposition: Proposition transportb_disp_cat_of_strict_twosided_disp_cat_mor_help {C₁ C₂ : setcategory} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {F G : C₁ ⟶ C₂} (p : F = G) (GG : twosided_disp_functor G G D₁ D₂) {x₁ x₂ y₁ y₂ : C₁} {f : x₁ --> x₂} {g : y₁ --> y₂} {xy₁ : D₁ x₁ y₁} {xy₂ : D₁ x₂ y₂} (fg : xy₁ -->[ f ][ g ] xy₂) : #2 (transportb (λ H, twosided_disp_functor H H D₁ D₂) p GG) fg = transportf_disp_mor2 (transportb_disp_cat_of_strict_twosided_disp_cat_mor_path p f) (transportb_disp_cat_of_strict_twosided_disp_cat_mor_path p g) (idtoiso_twosided_disp (idpath _) (idpath _) (transportb_disp_cat_of_strict_twosided_disp_cat_ob_help D₁ D₂ p GG xy₁) ;;2 transportb_disp_ob2_mor _ _ _ ;;2 #2 GG fg ;;2 transportb_disp_ob2_inv _ _ _ ;;2 idtoiso_twosided_disp (idpath _) (idpath _) (!transportb_disp_cat_of_strict_twosided_disp_cat_ob_help D₁ D₂ p GG xy₂)).
proof: Proof. induction p ; cbn. rewrite !(id_two_disp_left (D := D₂)). rewrite !(id_two_disp_right (D := D₂)). unfold transportb_disp_mor2. rewrite (two_disp_pre_whisker_f (D := D₂)). rewrite !(id_two_disp_left (D := D₂)). 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.PrecategoryBinProduct 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.DisplayedCatEq UniMath.CategoryTheory.DisplayedCats.Examples.DispFunctorPair 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