Definition opcleaving_weq_cleaving {C : category} (D : disp_cat C) : opcleaving D ≃ @cleaving _ (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 d' => total2 (fun ff => @is_cartesian _ _ _ _ _ f d' ff)) (pr1 ℓ) (tpair (@is_cartesian _ _ _ _ _ f (pr1 ℓ)) (pr12 ℓ) (pr1weq (@is_opcartesian_weq_is_cartesian _ D _ _ _ _ _ (pr12 ℓ)) (pr22 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_opcartesian_weq_is_cartesian _) (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.