Lemma disp_nat_trans_comp_inv {α: nat_z_iso F G} {α': nat_z_iso G H} (β: disp_nat_z_iso DF DG α) (β': disp_nat_z_iso DG DH α') (c:C) (d:D c) : is_disp_inverse (nat_z_iso_pointwise_z_iso (nat_z_iso_comp α α') c) (disp_nat_trans_comp β β' c d) (disp_nat_trans_comp (disp_nat_z_iso_inv β') (disp_nat_z_iso_inv β) c d). Proof. split. - etrans. apply assoc_disp. apply PartA.transportb_transpose_left. etrans. apply assoc4_disp. apply PartA.transportb_transpose_left. etrans. apply id_conjugation_disp. * apply disp_nat_z_iso_inv_iso. * apply disp_nat_z_iso_inv_iso. * unfold transportb. repeat rewrite transport_f_f. apply two_arg_paths. -- apply uip. apply homset_property. -- reflexivity. - etrans. apply assoc_disp. apply PartA.transportb_transpose_left. etrans. apply assoc4_disp. apply PartA.transportb_transpose_left. etrans. apply id_conjugation_disp. * apply disp_nat_z_iso_iso_inv. * apply disp_nat_z_iso_iso_inv. * unfold transportb. repeat rewrite transport_f_f. apply two_arg_paths. -- apply uip. apply homset_property. -- reflexivity. 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.