Lemma 73.4.10. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism in $(\textit{Spaces}/S)_{\acute{e}tale}$. The functor

\[ u : (\textit{Spaces}/Y)_{\acute{e}tale}\longrightarrow (\textit{Spaces}/X)_{\acute{e}tale}, \quad V/Y \longmapsto V/X \]

is cocontinuous, and has a continuous right adjoint

\[ v : (\textit{Spaces}/X)_{\acute{e}tale}\longrightarrow (\textit{Spaces}/Y)_{\acute{e}tale}, \quad (U \to X) \longmapsto (U \times _ X Y \to Y). \]

They induce the same morphism of topoi

\[ f_{big} : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \]

We have $f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. We have $f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times _ X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.

**Proof.**
The functor $u$ is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma 73.4.7). Hence Sites, Lemmas 7.21.5 and 7.21.6 apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/Y$ and $V/X$ we have $\mathop{\mathrm{Mor}}\nolimits _ X(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ Y(U, V \times _ X Y)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$.
$\square$

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