The existence of an fppf scheme U \to S and a 1-morphism of fibered categories \mathcal{U} \to \mathcal{X} which is surjective and smooth depends on defining a smooth and surjective morphisms of fibered categories. Here \mathcal{U} is the algebraic stack from the representable functor h_U on h_U: (Sch/S)_{fppf}^{op} \to Sets upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the seth_U(T) = \text{Hom}_{(Sch/S)_{fppf}}(T,U)is considered as a category, denoted h_\mathcal{U}(T), with objects in h_U(T) as fppf morphismsf:T \to Uand morphisms are the identity morphism. Henceh_{\mathcal{U}}:(Sch/S)_{fppf}^{op} \to Groupoidsis a 2-functor of groupoids. Showing this 2-functor is a sheaf is the content of the 2-Yoneda lemma. Using the Grothendieck construction, there is an associated category fibered in groupoids denoted \mathcal{U} \to \mathcal{X}.
