Lemma 64.7.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F \to H$, $G \to H$ be morphisms of algebraic spaces over $S$. Then $F \times _ H G$ is an algebraic space, and is a fibre product in the category of algebraic spaces over $S$.

**Proof.**
It follows from the stronger Lemma 64.7.2 that $F \times _ H G$ is an algebraic space. It is clear that $F \times _ H G$ is a fibre product in the category of algebraic spaces over $S$ since that is a full subcategory of the category of (pre)sheaves of sets on $(\mathit{Sch}/S)_{fppf}$.
$\square$

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