Lemma 20.42.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L, M$ be objects of $D(\mathcal{O}_ X)$. There is a canonical morphism

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

Lemma 20.42.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L, M$ be objects of $D(\mathcal{O}_ X)$. There is a canonical morphism

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

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

**Proof.**
Choose a K-injective complex $\mathcal{I}^\bullet $ representing $M$, a K-injective complex $\mathcal{J}^\bullet $ representing $L$, and a K-flat complex $\mathcal{K}^\bullet $ representing $K$. The map is defined using the map

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

of Lemma 20.41.5. By our particular choice of complexes the left hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _{\mathcal{O}_ X}^\mathbf {L} K$ and the right hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L), M)$. We omit the proof that this is functorial in all three objects of $D(\mathcal{O}_ X)$. $\square$

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