Let $X \to S$ be a projective flat morphism of finite presentation, $H$ the Hilbert scheme, and $W \subseteq X \times_S H$ the universal closed subscheme.
Then is the projection $ p : W \to H$ surjective?
In Bosch, Lutkebohmern, Raynaud's "Neron models", the authors say that "Due to the definition of $\operatorname{Hilb}_{X/S}$, the map $p$ is faithfully flat, proper, and finitely presented". But by the definition, we only have that $p$ is flat proper of finite presentation. Why is this $p$ surjective?
I want to use the fppf descent for a closed subscheme of $W$.
Thank you very much!