Lemma disp_functor_idtoiso_twosided_disp {x y : C} (xx := x ,, y : category_binproduct C C) {xy₁ xy₂ : D₁ x y} (p : xy₁ = xy₂) (f := idtoiso_twosided_disp (idpath x) (idpath y) p : xy₁ -->[ identity _ ][ identity _ ] xy₂) (φ := f : @mor_disp _ (twosided_disp_cat_to_disp_cat C C D₁) xx xx xy₁ xy₂ (identity _ ,, identity _)) (ψ := idtoiso_twosided_disp (idpath _) (idpath _) (maponpaths (F xx) p) : F xx xy₁ -->[ identity _ ][ identity _ ] F xx xy₂) : (♯F φ = ψ)%mor_disp. Proof. induction p ; cbn. exact (disp_functor_id F (x := xx) xy₁). 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.