Lemma 21.33.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}$-modules there is a canonical morphism

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

of complexes of $\mathcal{O}$-modules functorial in all three complexes.

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