Definition total_precategory_data_of_set_functor {C : setcategory} (F : C ⟶ cat_of_setcategory) : precategory_data. Proof. use make_precategory_data. - exact (total_precategory_ob_mor_of_set_functor F). - refine (λ x, identity _ ,, _). apply idtoiso. exact (eq_in_set_fiber (functor_id F (pr1 x)) (pr2 x)). - refine (λ x y z f g, pr1 f · pr1 g ,, _ · # (pr1 (# F (pr1 g))) (pr2 f) · pr2 g). apply idtoiso. exact (eq_in_set_fiber (functor_comp F (pr1 f) (pr1 g)) (pr2 x)). 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.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories.