Proposition transportb_disp_cat_of_strict_twosided_disp_cat_mor {C₁ C₂ : setcategory} {D₁ : strict_twosided_disp_cat C₁ C₁} {D₂ : strict_twosided_disp_cat C₂ C₂} (FD : functor_data C₁ C₂) (HF₁ HF₂ : is_functor FD) (F := make_functor FD HF₁) (G := make_functor FD HF₂) (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 (id_right _ @ id_left _) (id_right _ @ id_left _) (idtoiso_twosided_disp (idpath _) (idpath _) (transportb_disp_cat_of_strict_twosided_disp_cat_ob F HF₁ HF₂ p GG xy₁) ;;2 #2 GG fg ;;2 idtoiso_twosided_disp (idpath _) (idpath _) (!(transportb_disp_cat_of_strict_twosided_disp_cat_ob F HF₁ HF₂ p GG xy₂))). Proof. rewrite transportb_disp_cat_of_strict_twosided_disp_cat_mor_help. unfold transportb_disp_ob2_mor, transportb_disp_ob2_inv. etrans. { apply maponpaths. do 3 apply maponpaths_2. apply idtoiso_twosided_disp_concat_strict. } unfold transportb_disp_mor2. rewrite !two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite transport_f_f_disp_mor2. etrans. { do 2 apply maponpaths. apply idtoiso_twosided_disp_concat_strict. } unfold transportb_disp_mor2. rewrite !two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. use transportf_disp_mor2_eq. apply idtoiso_twosided_disp_triple_concat. 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.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.DispFunctorPair. 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. Section FromIso. End FromIso.