Definition cat_of_elems_on_nat_trans_data (α : X ⟹ Y) : functor_data (∫ X) (∫ Y). Proof. exists (λ a, (get_ob a,,α (get_ob a) (get_el a))). intros b c f. exists (get_mor f). abstract (exact (!eqtohomot (pr2 α (get_ob b) (get_ob c) (get_mor f)) (get_el b) @ maponpaths (α (get_ob c)) (get_eqn f))). Defined.
Require Import UniMath.Foundations.All. 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.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.