Definition fiber_equiv {D D' : disp_cat C} {FF : disp_functor (functor_identity _) D D'} (EFF : is_equiv_over_id FF) (c : C) : adj_equivalence_of_cats (fiber_functor FF c). Proof. exists (fiber_is_left_adj EFF c). destruct EFF as [[[GG [η ε]] tris] isos]; cbn in isos; cbn. use tpair. + intros d. apply is_z_iso_fiber_from_is_z_iso_disp. apply (is_z_iso_unit_over_id isos). + intros d. apply is_z_iso_fiber_from_is_z_iso_disp. apply (is_z_iso_counit_over_id isos). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber.