Definition disp_cat_from_preshv (D : PreShv C) : disp_cat C. Proof. use tpair. - use tpair. + exists (λ c, pr1hSet (pr1 D c)). intros x y c d f. exact (functor_on_morphisms (pr1 D) f d = c). + split. * intros; cbn in *; apply (toforallpaths _ _ _ (functor_id D x ) _ ). * intros ? ? ? ? ? ? ? ? X X0; cbn in *; etrans; [apply (toforallpaths _ _ _ (functor_comp D g f ) _ ) |]; cbn; etrans; [ apply maponpaths; apply X0 |]; apply X. - abstract ( repeat use tpair; cbn; intros; try apply setproperty; apply isasetaprop; apply setproperty ). 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.