Lemma 18.40.1. 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 12.26.6 and Lemma 18.28.7 we see that $g_!$ exists.
$\square$

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