Lemma 61.12.10. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. Let $\mathit{Sch}$ be the category of all schemes.

1. The categories $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ have fibre products agreeing with fibre products in $\mathit{Sch}$.

2. The categories $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$ have equalizers agreeing with equalizers in $\mathit{Sch}$.

3. The categories $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, and $S_{pro\text{-}\acute{e}tale}$ both have a final object, namely $S/S$.

4. The category $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ has a final object agreeing with the final object of $\mathit{Sch}$, namely $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

Proof. The category $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ contains $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and is closed under products and fibre products by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale})$. The fibre product $V \times _ U W$ in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ 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 $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. This proves the result for $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$. If $U \to S$, $V \to U$ and $W \to U$ are weakly étale then so is $V \times _ U W \to S$ (see More on Morphisms, Section 37.61) and hence we get fibre products for $S_{pro\text{-}\acute{e}tale}$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence we get fibre products for $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$.

Let $a, b : U \to V$ be two morphisms in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is

$V \times _{\Delta _{V/\mathop{\mathrm{Spec}}(\mathbf{Z})}, V \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} V, (a, b)} (U \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} U)$

which is an object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ by what we saw above. Thus $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ has equalizers. If $a$ and $b$ are morphisms over $S$, then the equalizer (in the category of schemes) is also given by

$V \times _{\Delta _{V/S}, V \times _ S V, (a, b)} (U \times _ S U)$

hence we see that $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ has equalizers. Moreover, if $U$ and $V$ are weakly-étale over $S$, then so is the equalizer above as a fibre product of schemes weakly étale over $S$. Thus $S_{pro\text{-}\acute{e}tale}$ has equalizers. The statements on final objects is clear. $\square$

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