Definition z_iso_disp_weq_z_iso_disp_reindex {C₁ C₂ : category} {F : C₁ ⟶ C₂} {D : disp_cat C₂} {x : C₁} (xx yy : D (F x)) : z_iso_disp (identity_z_iso (F x)) xx yy ≃ @z_iso_disp _ (reindex_disp_cat F D) _ _ (identity_z_iso x) xx yy. Proof. use make_weq. - exact z_iso_disp_to_z_iso_disp_reindex. - use isweq_iso. + exact z_iso_disp_reindex_to_z_iso_disp. + abstract (intros i ; use subtypePath ; [ intro ; apply isaprop_is_z_iso_disp | ] ; cbn ; apply transportfbinv). + abstract (intros i ; use subtypePath ; [ intro ; apply isaprop_is_z_iso_disp | ] ; cbn ; apply transportbfinv). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Section IsCartesianFromReindexDispCat. End IsCartesianFromReindexDispCat. Section IsOpCartesianFromReindexDispCat. End IsOpCartesianFromReindexDispCat.