Lemma 66.16.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ is reduced. Then

the scheme theoretic image $Z$ of $f$ is the reduced induced algebraic space structure on $\overline{|f|(|X|)}$, and

for any étale morphism $V \to Y$ the scheme theoretic image of $X \times _ Y V \to V$ is equal to $Z \times _ Y V$.

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