Lemma 38.34.14. Let $S$ be a scheme. Let $\mathit{Sch}_ h$ be a big h site containing $S$. The underlying categories of the sites $\mathit{Sch}_ h$, $(\mathit{Sch}/S)_ h$, and $(\textit{Aff}/S)_ h$ have fibre products. In each case the obvious functor into the category $\mathit{Sch}$ of all schemes commutes with taking fibre products. The category $(\mathit{Sch}/S)_ h$ has a final object, namely $S/S$.

Proof. For $\mathit{Sch}_ h$ 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 $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_ h)$. The fibre product $V \times _ U W$ in $\mathit{Sch}_ h$ 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 $(\mathit{Sch}/S)_ h$. This proves the result for $(\mathit{Sch}/S)_ h$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence the result for $(\textit{Aff}/S)_ h$. $\square$

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