Lemma 59.87.5. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation with geometrically reduced fibres. Then $f^{-1} : \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ commutes with products.
Proof. Let $I$ be a set and let $\mathcal{G}_ i$ be a sheaf on $S_{\acute{e}tale}$ for $i \in I$. Let $U \to X$ be an étale morphism such that $U \to S$ factors as $U \to V \to S$ with $V \to S$ étale and $U \to V$ flat of finite presentation with geometrically connected fibres. Then we have
\begin{align*} f^{-1}(\prod \mathcal{G}_ i)(U) & = (\prod \mathcal{G}_ i)(V) \\ & = \prod \mathcal{G}_ i(V) \\ & = \prod f^{-1}\mathcal{G}_ i(U) \\ & = (\prod f^{-1}\mathcal{G}_ i)(U) \end{align*}
where we have used Lemma 59.39.3 in the first and third equality (we are also using that the restriction of $f^{-1}\mathcal{G}$ to $U_{\acute{e}tale}$ is the pullback via $U \to V$ of the restriction of $\mathcal{G}$ to $V_{\acute{e}tale}$, see Sites, Lemma 7.28.2). By More on Morphisms, Lemma 37.46.3 every object $U$ of $X_{\acute{e}tale}$ has an étale covering $\{ U_ i \to U\} $ such that the discussion in the previous paragraph applies to $U_ i$. The lemma follows. $\square$
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