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 $D(\mathcal{A}'', \text{d}) \to D(\mathcal{A}, \text{d})$.

Proof. Follows from Lemmas 24.28.7 and 24.29.3 and uniqueness of adjoints. $\square$

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