Lemma form_equiv_inv_adjunction_data : form_equiv_over_id inv_adjunction_data. Proof. cbn. use tpair. + intros. cbn. set (XR:= @is_z_iso_inv_from_is_z_iso_disp). specialize (XR _ D _ _ _ _ _ _ (is_z_iso_counit_over_id (pr2 isEquiv) x xx)). cbn in XR. eapply is_z_iso_disp_independent_of_is_z_iso. apply XR. + cbn. intros. set (XR:= @is_z_iso_inv_from_is_z_iso_disp). specialize (XR _ D' _ _ _ _ _ _ (is_z_iso_unit_over_id (pr2 isEquiv) x xx)). cbn in XR. eapply is_z_iso_disp_independent_of_is_z_iso. apply XR. Qed.
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.