Lemma 61.31.3. Suppose given big sites $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ and $\mathit{Sch}'_{pro\text{-}\acute{e}tale}$ as in Definition 61.12.7. Assume that $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ is contained in $\mathit{Sch}'_{pro\text{-}\acute{e}tale}$. The inclusion functor $\mathit{Sch}_{pro\text{-}\acute{e}tale}\to \mathit{Sch}'_{pro\text{-}\acute{e}tale}$ satisfies the assumptions of Sites, Lemma 7.21.8. There are morphisms of topoi
\begin{eqnarray*} g : \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}'_{pro\text{-}\acute{e}tale}) \\ f : \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}'_{pro\text{-}\acute{e}tale}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale}) \end{eqnarray*}
such that $f \circ g \cong \text{id}$. For any object $S$ of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ the inclusion functor $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$ satisfies the assumptions of Sites, Lemma 7.21.8 also. Hence similarly we obtain morphisms
\begin{eqnarray*} g : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}) \\ f : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) \end{eqnarray*}
with $f \circ g \cong \text{id}$.
Proof.
Assumptions (b), (c), and (e) of Sites, Lemma 7.21.8 are immediate for the functors $\mathit{Sch}_{pro\text{-}\acute{e}tale}\to \mathit{Sch}'_{pro\text{-}\acute{e}tale}$ and $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$. Property (a) holds by Lemma 61.31.1. Property (d) holds because fibre products in the categories $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $\mathit{Sch}'_{pro\text{-}\acute{e}tale}$ exist and are compatible with fibre products in the category of schemes.
$\square$
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