Lemma 38.34.17. Let $\mathit{Sch}_ h$ be a big h site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_ h$. The functor

$u : (\mathit{Sch}/T)_ h \longrightarrow (\mathit{Sch}/S)_ h, \quad V/T \longmapsto V/S$

is cocontinuous, and has a continuous right adjoint

$v : (\mathit{Sch}/S)_ h \longrightarrow (\mathit{Sch}/T)_ h, \quad (U \to S) \longmapsto (U \times _ S T \to T).$

They induce the same morphism of topoi

$f_{big} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/T)_ h) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_ h)$

We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T)$. 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. 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 $U/T$ and $V/S$ we have $\mathop{\mathrm{Mor}}\nolimits _ S(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ T(U, V \times _ S T)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).