Lemma 34.4.10. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$. The underlying categories of the sites $\mathit{Sch}_{\acute{e}tale}$, $(\mathit{Sch}/S)_{\acute{e}tale}$, $S_{\acute{e}tale}$, and $(\textit{Aff}/S)_{\acute{e}tale}$ have fibre products. In each case the obvious functor into the category $\mathit{Sch}$ of all schemes commutes with taking fibre products. The categories $(\mathit{Sch}/S)_{\acute{e}tale}$, and $S_{\acute{e}tale}$ both have a final object, namely $S/S$.

Proof. For $\mathit{Sch}_{\acute{e}tale}$ it is true by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{\acute{e}tale})$. The fibre product $V \times _ U W$ in $\mathit{Sch}_{\acute{e}tale}$ is a fibre product in $\mathit{Sch}$ and is the fibre product of $V/S$ with $W/S$ over $U/S$ in the category of all schemes over $S$, and hence also a fibre product in $(\mathit{Sch}/S)_{\acute{e}tale}$. This proves the result for $(\mathit{Sch}/S)_{\acute{e}tale}$. If $U \to S$, $V \to U$ and $W \to U$ are étale then so is $V \times _ U W \to S$ and hence we get the result for $S_{\acute{e}tale}$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence the result for $(\textit{Aff}/S)_{\acute{e}tale}$. $\square$

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