Definition cleaving_reindex_disp_cat {C₁ C₂ : category} (F : C₁ ⟶ C₂) (D : disp_cat C₂) (HD : cleaving D) : cleaving (reindex_disp_cat F D). Proof. intros x y f d. pose (HD (F x) (F y) (#F f) d) as lift. simple refine (_ ,, (_ ,, _)). - exact (pr1 lift). - exact (pr12 lift). - simpl. apply is_cartesian_in_reindex_disp_cat. exact (pr22 lift). 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.