Lemma 64.7.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G$ be algebraic spaces over $S$. Then $F \times G$ is an algebraic space, and is a product in the category of algebraic spaces over $S$.

**Proof.**
It is clear that $H = F \times G$ is a sheaf. The diagonal of $H$ is simply the product of the diagonals of $F$ and $G$. Hence it is representable by Lemma 64.3.4. Finally, if $U \to F$ and $V \to G$ are surjective étale morphisms, with $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, then $U \times V \to F \times G$ is surjective étale by Lemma 64.5.7.
$\square$

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