Lemma 6.24.7. Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $Y$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{F}$ be a sheaf of $f^{-1}\mathcal{O}$-modules. Then

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \]

Here we use Lemmas 6.24.6 and 6.24.5, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$-module by restriction via $\mathcal{O} \to f_*f^{-1}\mathcal{O}$.

**Proof.**
Argue by the equalities

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

where the second is Lemmas 6.24.3 and the first is by Lemma 6.20.1.
$\square$

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