Definition functor_total_category_of_set_functor_data : functor_data (total_setcategory_of_set_functor G₁) (total_setcategory_of_set_functor G₂). Proof. use make_functor_data. - exact (λ x, F (pr1 x) ,, pr1 (pr1 α (pr1 x)) (pr2 x)). - refine (λ x y f, #F (pr1 f) ,, _). exact (idtoiso (functor_total_category_of_set_functor_eq f) · # (pr1 (α (pr1 y))) (pr2 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.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.GrothendieckConstruction.TotalCategory. Section FunctorTotalCategoryFromSetFunctor. End FunctorTotalCategoryFromSetFunctor. Section NatTransTotalCategoryFromNatTrans. End NatTransTotalCategoryFromNatTrans.