Lemma 18.41.1. Let u : \mathcal{C} \to \mathcal{D} be a continuous and cocontinuous functor between sites. Denote g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) the associated morphism of topoi. Let \mathcal{O}_\mathcal {D} be a sheaf of rings on \mathcal{D}. Set \mathcal{O}_\mathcal {C} = g^{-1}\mathcal{O}_\mathcal {D}. Hence g becomes a morphism of ringed topoi with g^* = g^{-1}. In this case there exists a functor
g_! : \textit{Mod}(\mathcal{O}_\mathcal {C}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal {D})
which is left adjoint to g^*.
Proof.
Let U be an object of \mathcal{C}. For any \mathcal{O}_\mathcal {D}-module \mathcal{G} we have
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(j_{U!}\mathcal{O}_ U, g^{-1}\mathcal{G}) & = g^{-1}\mathcal{G}(U) \\ & = \mathcal{G}(u(U)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(j_{u(U)!}\mathcal{O}_{u(U)}, \mathcal{G}) \end{align*}
because g^{-1} is described by restriction, see Sites, Lemma 7.21.5. Of course a similar formula holds a direct sum of modules of the form j_{U!}\mathcal{O}_ U. By Homology, Lemma 12.29.6 and Lemma 18.28.8 we see that g_! exists.
\square
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