Lemma 34.3.13. Let $\mathit{Sch}_{Zar}$ be a big Zariski site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{Zar}$. The functor $T_{Zar} \to (\mathit{Sch}/S)_{Zar}$ is cocontinuous and induces a morphism of topoi
\[ i_ f : \mathop{\mathit{Sh}}\nolimits (T_{Zar}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar}) \]
For a sheaf $\mathcal{G}$ on $(\mathit{Sch}/S)_{Zar}$ we have the formula $(i_ f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_ f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.
Proof.
Denote the functor $u : T_{Zar} \to (\mathit{Sch}/S)_{Zar}$. In other words, given and open immersion $j : U \to T$ corresponding to an object of $T_{Zar}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma 34.3.9. Moreover, $T_{Zar}$ has equalizers (as any two morphisms with the same source and target are the same) and $u$ commutes with them. It is clearly cocontinuous. It is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the lemma follows from Sites, Lemmas 7.21.5 and 7.21.6.
$\square$
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