Definition nat_trans_total_category_of_set_functor_data : nat_trans_data (functor_total_category_of_set_functor F₁ β₁) (functor_total_category_of_set_functor F₂ β₂). Proof. refine (λ x, α (pr1 x) ,, _). exact (idtoiso (nat_trans_total_category_of_set_functor_eq 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.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.