Lemma pointwise_inverse_disp_nat_trans_over_id_right {C : category} {D : disp_cat C} {D' : disp_cat C} {FF GG : disp_functor (functor_identity _) D D'} (αα : disp_nat_trans (nat_trans_id _) FF GG) (Hαα : ∏ (x : C) (xx : D x), is_z_iso_disp (identity_z_iso x) (pr1 αα x xx)) : disp_nat_trans_over_id_comp (pointwise_inverse_disp_nat_trans αα Hαα) αα = disp_nat_trans_id _. Proof. use disp_nat_trans_eq. intros x xx ; cbn. etrans. { apply maponpaths. apply (z_iso_disp_after_inv_mor (Hαα x xx)). } unfold transportb. rewrite transport_f_f. apply transportf_set. apply homset_property. Qed.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. 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.whiskering. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Isos.