The Stacks project

Lemma 80.3.8. Let $S$ be a scheme. Let

\[ \xymatrix{ F \ar[r]^ a & G \ar[r]^ b & H } \]

be maps of presheaves on $(\mathit{Sch}/S)_{fppf}$. If $a$ and $b$ are representable by algebraic spaces, so is $b \circ a$.

Proof. Let $T$ be a scheme over $S$, and let $T \to H$ be a morphism. By assumption $T \times _ H G$ is an algebraic space. Hence by Lemma 80.3.7 we see that $T \times _ H F = (T \times _ H G) \times _ G F$ is an algebraic space as well. $\square$

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