Definition z_iso_disp_reindex_to_z_iso_disp {C₁ C₂ : category} {F : C₁ ⟶ C₂} {D : disp_cat C₂} {x : C₁} {xx yy : D (F x)} : @z_iso_disp _ (reindex_disp_cat F D) _ _ (identity_z_iso x) xx yy → z_iso_disp (identity_z_iso (F x)) xx yy. Proof. intros i. use make_z_iso_disp. - exact (transportf (λ z, _ -->[ z ] _) (functor_id _ _) i). - simple refine (_ ,, _ ,, _). + exact (transportf (λ z, _ -->[ z ] _) (functor_id _ _) (inv_mor_disp_from_z_iso i)). + abstract (unfold transportb ; rewrite mor_disp_transportf_prewhisker ; rewrite mor_disp_transportf_postwhisker ; rewrite !transport_f_f ; etrans ; [ apply maponpaths ; pose (z_iso_disp_after_inv_mor i) as p ; cbn in p ; exact (transportf_transpose_right p) | ] ; unfold transportb ; rewrite !transport_f_f ; etrans ; [ apply maponpaths ; apply transportf_reindex | ] ; rewrite !transport_f_f ; apply maponpaths_2 ; apply homset_property). + abstract (unfold transportb ; rewrite mor_disp_transportf_prewhisker ; rewrite mor_disp_transportf_postwhisker ; rewrite !transport_f_f ; etrans ; [ apply maponpaths ; pose (inv_mor_after_z_iso_disp i) as p ; cbn in p ; exact (transportf_transpose_right p) | ] ; unfold transportb ; rewrite !transport_f_f ; etrans ; [ apply maponpaths ; apply transportf_reindex | ] ; rewrite !transport_f_f ; apply maponpaths_2 ; apply homset_property). 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.