Definition is_cartesian_id_disp {C : category} {D : disp_cat C} {x : C} (xx : D x) : is_cartesian (id_disp xx). Proof. intros z g zz hh. use iscontraprop1. - abstract (use invproofirrelevance ; intros f₁ f₂ ; use subtypePath ; [ intro ; intros ; apply D | ] ; refine (id_right_disp_var _ @ _ @ !(id_right_disp_var _)) ; rewrite (pr2 f₁), (pr2 f₂) ; apply idpath). - use tpair. + exact (transportf _ (id_right _) hh). + abstract (simpl ; rewrite id_right_disp ; unfold transportb ; rewrite transport_f_f ; rewrite pathsinv0r ; apply idpath). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Opposite. Section IsoCleavingFromOpcleaving. End IsoCleavingFromOpcleaving.