Lemma 65.5.7. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F_ i, G_ i : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$, $i = 1, 2$. Let $a_ i : F_ i \to G_ i$, $i = 1, 2$ be representable transformations of functors. Let $\mathcal{P}$ be a property as in Definition 65.5.1 which is stable under composition. If $a_1$ and $a_2$ have property $\mathcal{P}$ so does $a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2$.

**Proof.**
Note that the lemma makes sense by Lemma 65.3.4. Proof omitted.
$\square$

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