Lemma 77.3.7. Let $S$ be a scheme. Let $a : F \to G$ be a map of presheaves on $(\mathit{Sch}/S)_{fppf}$. Suppose $a : F \to G$ is representable by algebraic spaces. If $X$ is an algebraic space over $S$, and $X \to G$ is a map of presheaves then $X \times _ G F$ is an algebraic space.

**Proof.**
By Lemma 77.3.3 the transformation $X \times _ G F \to X$ is representable by algebraic spaces. Hence it is an algebraic space by Lemma 77.3.6.
$\square$

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