Lemma 21.34.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\mathcal{O})$ there is a canonical morphism

in $D(\mathcal{O})$ functorial in $K, L, M$.

Lemma 21.34.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\mathcal{O})$ there is a canonical morphism

\[ K \otimes _\mathcal {O}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K \otimes _\mathcal {O}^\mathbf {L} L) \]

in $D(\mathcal{O})$ functorial in $K, L, M$.

**Proof.**
Choose a K-flat complex $\mathcal{K}^\bullet $ representing $K$, and a K-injective complex $\mathcal{I}^\bullet $ representing $L$, and choose any complex of $\mathcal{O}$-modules $\mathcal{M}^\bullet $ representing $M$. Choose a quasi-isomorphism $\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{I}^\bullet ) \to \mathcal{J}^\bullet $ where $\mathcal{J}^\bullet $ is K-injective. Then we use the map

\[ \text{Tot}\left( \mathcal{K}^\bullet \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{I}^\bullet ) \right) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{I}^\bullet )) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{J}^\bullet ) \]

where the first map is the map from Lemma 21.33.4. $\square$

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