The Stacks project

Lemma 80.3.10. 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

  1. $\Delta : G \to G \times _ H G$ is representable by algebraic spaces, and

  2. $b \circ a : F \to H$ is representable by algebraic spaces.

Then $a$ is representable by algebraic spaces.

Proof. Let $U$ be a scheme over $S$ and let $\xi \in G(U)$. Then

\[ U \times _{\xi , G, a} F = (U \times _{b(\xi ), H, b \circ a} F) \times _{(\xi , a), (G \times _ H G), \Delta } G \]

Hence the result using Lemma 80.3.7. $\square$

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