The Stacks project

Lemma 80.4.3. Let $S$ be a scheme. Let $\mathcal{P}$ be a property as in Definition 80.4.1, and assume $\mathcal{P}$ is stable under composition. Let

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

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

Proof. Omitted. Hint: See Lemma 80.3.8 and use stability under composition. $\square$

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