Lemma 65.3.4. 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. Then

\[ a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2 \]

is a representable transformation of functors.

**Proof.**
Write $a_1 \times a_2$ as the composition $F_1 \times F_2 \to G_1 \times F_2 \to G_1 \times G_2$. The first arrow is the base change of $a_1$ by the map $G_1 \times F_2 \to G_1$, and the second arrow is the base change of $a_2$ by the map $G_1 \times G_2 \to G_2$. Hence this lemma is a formal consequence of Lemmas 65.3.2 and 65.3.3.
$\square$

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