Lemma is_functor_fiber_functor : is_functor fiber_functor_data. Proof. split; unfold functor_idax, functor_compax; cbn. - intros. apply transportf_pathsinv0. apply pathsinv0. apply disp_functor_id. - intros. etrans. apply maponpaths. apply disp_functor_transportf. etrans. apply transport_f_f. etrans. apply maponpaths. apply disp_functor_comp. etrans. apply transport_f_f. apply pathsinv0. etrans. apply maponpaths. apply mor_disp_transportf_prewhisker. etrans. apply transport_f_f. etrans. apply maponpaths. apply mor_disp_transportf_postwhisker. etrans. apply transport_f_f. apply maponpaths_2, homset_property. Qed.
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.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Total.