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 (maponpaths (α (get_ob b)) (get_eqn f) @ eqtohomot (pr2 α (get_ob c) (get_ob b) (get_mor f)) (get_el c))). Defined.
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.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Presheaf.