Lemma 80.11.4. Let $S$ be a scheme. Let $a : F \to G$ and $b : G \to H$ be transformations of functors $(\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Assume
$F, G, H$ are sheaves,
$a : F \to G$ is representable by algebraic spaces, flat, locally of finite presentation, and surjective, and
$b \circ a : F \to H$ is representable by algebraic spaces.
Then $b$ is representable by algebraic spaces.
Proof. Let $U$ be a scheme over $S$ and let $\xi \in H(U)$. We have to show that $U \times _{\xi , H} G$ is an algebraic space. On the other hand, we know that $U \times _{\xi , H} F$ is an algebraic space and that $U \times _{\xi , H} F \to U \times _{\xi , H} G$ is representable by algebraic spaces, flat, locally of finite presentation, and surjective as a base change of the morphism $a$ (see Lemma 80.4.2). Thus the result follows from Theorem 80.10.1. $\square$
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