Lemma 24.29.7. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{A}, \mathcal{A}', \mathcal{A}'' be differential graded \mathcal{O}-algebras. Let \mathcal{N} and \mathcal{N}' be a differential graded (\mathcal{A}, \mathcal{A}')-bimodule and (\mathcal{A}', \mathcal{A}'')-bimodule. Assume that the canonical map
\mathcal{N} \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' \longrightarrow \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}'
in D(\mathcal{A}'', \text{d}) is a quasi-isomorphism. Then we have
R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{A}''} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}', -) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{A}'}(\mathcal{N}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{A}''}(\mathcal{N}', -))
as functors D(\mathcal{A}'', \text{d}) \to D(\mathcal{A}, \text{d}).
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