Lemma 79.11.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \to Y$ and $g : X \to Z$ be morphisms of algebraic spaces over $B$. Assume
$Y \to B$ is separated,
$g$ is surjective, flat, and locally of finite presentation,
there is a scheme theoretically dense open $V \subset Z$ such that $f|_{g^{-1}(V)} : g^{-1}(V) \to Y$ factors through $V$.
Then $f$ factors through $g$.
Proof. Set $R = X \times _ Z X$. By (2) we see that $Z = X/R$ as sheaves. Also (2) implies that the inverse image of $V$ in $R$ is scheme theoretically dense in $R$ (Morphisms of Spaces, Lemma 67.30.11). The we see that the two compositions $R \to X \to Y$ are equal by Morphisms of Spaces, Lemma 67.17.8. The lemma follows. $\square$
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