Lemma 22.33.4. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $N$ be a differential graded $(A, B)$-bimodule which has property (P) as a left differential graded $A$-module. Then $M \otimes _ A^\mathbf {L} N$ is computed by $M \otimes _ A N$ for all differential graded $A$-modules $M$.

**Proof.**
Let $f : M \to M'$ be a homomorphism of differential graded $A$-modules which is a quasi-isomorphism. We claim that $f \otimes \text{id} : M \otimes _ A N \to M' \otimes _ A N$ is a quasi-isomorphism. If this is true, then by the construction of the derived tensor product in the proof of Lemma 22.33.2 we obtain the desired result. The construction of the map $f \otimes \text{id}$ only depends on the left differential graded $A$-module structure on $N$. Moreover, we have $M \otimes _ A N = N \otimes _{A^{opp}} M = N \otimes _{A^{opp}}^\mathbf {L} M$ because $N$ has property (P) as a differential graded $A^{opp}$-module. Hence the claim follows from Lemma 22.33.3.
$\square$

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