Lemma 66.12.4. Let S be a scheme. Let X, Y be algebraic spaces over S. Let Z \subset X be a closed subspace. Assume Y is reduced. A morphism f : Y \to X factors through Z if and only if f(|Y|) \subset |Z|.
Proof. Assume f(|Y|) \subset |Z|. Choose a diagram
\xymatrix{ V \ar[d]_ b \ar[r]_ h & U \ar[d]^ a \\ Y \ar[r]^ f & X }
where U, V are schemes, and the vertical arrows are surjective and étale. The scheme V is reduced, see Lemma 66.7.1. Hence h factors through a^{-1}(Z) by Schemes, Lemma 26.12.7. So a \circ h factors through Z. As Z \subset X is a subsheaf, and V \to Y is a surjection of sheaves on (\mathit{Sch}/S)_{fppf} we conclude that X \to Y factors through Z. \square
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