Lemma 21.20.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. For $L$ in $D(\mathcal{O}_ U)$ and $K$ in $D(\mathcal{O})$ we have $j_!L \otimes _\mathcal {O}^\mathbf {L} K = j_!(L \otimes _{\mathcal{O}_ U}^\mathbf {L} K|_ U)$.

**Proof.**
Represent $L$ by a complex of $\mathcal{O}_ U$-modules and $K$ by a K-flat complexe of $\mathcal{O}$-modules and apply Modules on Sites, Lemma 18.27.7. Details omitted.
$\square$

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