Definition iterated_slice_equivalence_of_cats : equivalence_of_cats (C/y/xf) (C/x). Proof. use make_equivalence_of_cats. - use make_adjunction_data. + exact from_iterated_slice. + exact to_iterated_slice. + exact iterated_slice_unit. + exact iterated_slice_counit. - use make_forms_equivalence. + intro h. use is_z_iso_fiber_from_is_z_iso_disp. use is_z_iso_disp_codomain. use is_z_iso_fiber_from_is_z_iso_disp. use is_z_iso_disp_codomain. cbn. apply is_z_isomorphism_identity. + intro h. use is_z_iso_fiber_from_is_z_iso_disp. use is_z_iso_disp_codomain. cbn. apply is_z_isomorphism_identity. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.PreservationProperties. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodColimits.