Proof.
Assume (1), and let Z \to Y be as in (2). Choose a scheme V and a surjective étale morphism V \to Y. By assumption the morphism of schemes V \times _ Y X \to V is universally closed. By Properties of Spaces, Section 66.4 in the commutative diagram
\xymatrix{ |V \times _ Y X| \ar[r] \ar[d] & |Z \times _ Y X| \ar[d] \\ |V| \ar[r] & |Z| }
the horizontal arrows are open and surjective, and moreover
|V \times _ Y X| \longrightarrow |V| \times _{|Z|} |Z \times _ Y X|
is surjective. Hence as the left vertical arrow is closed it follows that the right vertical arrow is closed. This proves (2). The implication (2) \Rightarrow (1) is immediate from the definitions.
\square
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