The Stacks project

Lemma 20.41.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given complexes $\mathcal{K}^\bullet , \mathcal{L}^\bullet , \mathcal{M}^\bullet $ of $\mathcal{O}_ X$-modules there is a canonical morphism

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

of complexes of $\mathcal{O}_ X$-modules.

Proof. Omitted. Hint: This is proved in exactly the same way as More on Algebra, Lemma 15.71.3. $\square$

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