Lemma is_disp_nat_trans_pointwise_inv (x y : FunctorsC'C) (f : z_iso x y) (xx : disp_functor_cat x) (yy : disp_functor_cat y) (FF : xx -->[ f] yy) (H : ∏ (x' : C') (xx' : D' x'), is_z_iso_disp (pointwise_z_iso_from_nat_z_iso f x') (pr1 FF x' xx')) : disp_nat_trans_axioms (λ x' xx', inv_mor_disp_from_z_iso (H x' xx')). Proof. intros x' x0 f0 xx' xx0 ff. apply (precomp_with_z_iso_disp_is_inj (make_z_iso_disp _ (H x' xx'))). refine (assoc_disp _ _ _ @ _). symmetry. refine (mor_disp_transportf_prewhisker _ _ _ @ _). refine (maponpaths (transportf _ _) (assoc_disp _ _ _) @ _). refine (transport_f_f _ _ _ _ @ _). refine (maponpaths (λ x, transportf _ _ (x ;; _)) (inv_mor_after_z_iso_disp (H _ _)) @ _). refine (maponpaths (transportf _ _) (mor_disp_transportf_postwhisker _ _ _) @ _). refine (maponpaths (λ x, transportf _ _ (transportf _ _ x)) (id_left_disp _) @ _). refine (maponpaths (transportf _ _) (transport_f_f _ _ _ _) @ _). refine (transport_f_f _ _ _ _ @ _). symmetry. apply transportf_transpose_left. refine (!maponpaths (λ x, x ;; _) (transportf_transpose_left (pr2 FF x' x0 f0 xx' xx0 ff)) @ _). refine (mor_disp_transportf_postwhisker _ _ _ @ _). refine (maponpaths (transportf _ _) (assoc_disp_var _ _ _) @ _). refine (transport_f_f _ _ _ _ @ _). refine (maponpaths (λ x, transportf _ _ (_ ;; x)) (inv_mor_after_z_iso_disp (H _ _)) @ _). refine (maponpaths (transportf _ _) (mor_disp_transportf_prewhisker _ _ _) @ _). refine (transport_f_f _ _ _ _ @ _). refine (maponpaths (λ x, transportf _ _ x) (id_right_disp _) @ _). refine (transport_f_f _ _ _ _ @ _). symmetry. refine (transport_f_f _ _ _ _ @ _). apply (maponpaths (λ x, transportf _ x _)). apply homset_property. Qed.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos.