Definition total_adjunction_forms_equivalence2_disp_inv : ∏ x : total_category D2, pr2 (functor_identity (total_category D2) x) -->[ inv_from_z_iso (pr1 (adjcounit (total_adjunction FF) x),, total_adjunction_forms_equivalence2_base x)] pr2 ((pr121 (total_adjunction FF) ∙ pr11 (total_adjunction FF)) x). Proof. intro x. set (f := pr1 (pr22 FF (pr1 x) (pr2 x))). cbn. assert (p : identity (pr1 x) · Isos.is_z_isomorphism_mor ((pr222 F) (pr1 x)) · identity (F (right_adjoint F (pr1 x))) = Isos.is_z_isomorphism_mor ((pr222 F) (pr1 x))). { rewrite id_left. now apply id_right. } exact (transportf (mor_disp _ _) (! p) f). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. 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.Adjunctions. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Isos.