The Stacks project

Lemma 18.13.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi or ringed sites. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {C}$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal {D}$-modules. There is a canonical bijection

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, f_*\mathcal{F}). \]

In other words: the functor $f^*$ is the left adjoint to $f_*$.

Proof. This follows from the work we did before:

\begin{eqnarray*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{G}, \mathcal{F}) & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}_\mathcal {C})}( \mathcal{O}_\mathcal {C} \otimes _{f^{-1}\mathcal{O}_\mathcal {D}} f^{-1}\mathcal{G}, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(f^{-1}\mathcal{O}_\mathcal {D})}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}\mathcal{O}_\mathcal {D}}) \\ & = & \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*}

Here we use Lemmas 18.11.3 and 18.12.3. $\square$

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