Lemma 18.27.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. For $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_ U)$ and $\mathcal{F}$ in $\textit{Mod}(\mathcal{O})$ we have $j_{U!}\mathcal{G} \otimes _\mathcal {O} \mathcal{F} = j_{U!}(\mathcal{G} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U)$.
Proof. Let $\mathcal{H}$ be an object of $\textit{Mod}(\mathcal{O})$. Then
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_{U!}(\mathcal{G} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U), \mathcal{H}) & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U, \mathcal{H}|_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_ U, \mathcal{H}|_ U)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{H})|_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_{U!}\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{H})) \\ & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_{U!}\mathcal{G} \otimes _\mathcal {O} \mathcal{F}, \mathcal{H}) \end{align*}
The first equality because $j_{U!}$ is a left adjoint to restriction of modules. The second by Lemma 18.27.6. The third by Lemma 18.27.2. The fourth because $j_{U!}$ is a left adjoint to restriction of modules. The fifth by Lemma 18.27.6. The lemma follows from this and the Yoneda lemma. $\square$
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