Definition fiber_nat_trans {C C' : category} {F : functor C C'} {D D'} {FF FF' : disp_functor F D D'} (α : disp_nat_trans (nat_trans_id F) FF FF') (c : C) : nat_trans (fiber_functor FF c) (fiber_functor FF' c). Proof. use tpair; simpl. - intro d. exact (α c d). - unfold is_nat_trans; intros d d' ff; simpl. set (αff := pr2 α _ _ _ _ _ ff); simpl in αff. cbn. etrans. apply maponpaths, mor_disp_transportf_postwhisker. etrans. apply transport_f_f. etrans. apply maponpaths, αff. etrans. apply transport_f_b. apply @pathsinv0. etrans. apply maponpaths, mor_disp_transportf_prewhisker. etrans. apply transport_f_f. apply maponpaths_2, homset_property. 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.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.