Definition total_adjunction_forms_equivalence1_disp_inv : ∏ x : total_category D1, RR (F (pr1 x)) (LL (pr1 x) (pr2 x)) -->[identity (R (F (pr1 x))) · is_z_isomorphism_mor ((pr122 F) (pr1 x)) · identity (pr1 x)] pr2 x. Proof. intro x. set (f := pr1 (pr12 FF (pr1 x) (pr2 x))). assert (p : identity (right_adjoint F (F (pr1 x))) · Isos.is_z_isomorphism_mor ((pr122 F) (pr1 x)) · identity (pr1 x) = Isos.is_z_isomorphism_mor ((pr122 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.