Definition cleaving_weq_opcleaving {C : category} (D : disp_cat C) : cleaving D ≃ @opcleaving _ (op_disp_cat D). Proof. use make_weq. - exact (λ HD c₁ c₂ cc₁ f, let ℓ := HD c₁ c₂ f cc₁ in tpair (fun cc₂ => total2 (fun ff => @is_opcartesian _ _ _ _ _ cc₁ cc₂ ff)) (pr1 ℓ) (tpair (@is_opcartesian _ _ _ _ _ cc₁ ℓ) (pr12 ℓ) (pr1weq (is_cartesian_weq_is_opcartesian ℓ) ℓ))). - use isweq_iso. + refine (λ HD c₁ c₂ cc₁ f, let ℓ := HD c₁ c₂ f cc₁ in pr1 ℓ ,, pr12 ℓ ,, _). exact (invmap (is_cartesian_weq_is_opcartesian _) (pr22 ℓ)). + intro ; apply idpath. + intro ; 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.