Lemma 22.33.5. 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. Then the functor

of Lemma 22.33.2 is a left adjoint to the functor

of Lemma 22.31.2.

Lemma 22.33.5. 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. Then the functor

\[ - \otimes _ A^\mathbf {L} N : D(A, \text{d}) \longrightarrow D(B, \text{d}) \]

of Lemma 22.33.2 is a left adjoint to the functor

\[ R\mathop{\mathrm{Hom}}\nolimits (N, -) : D(B, \text{d}) \longrightarrow D(A, \text{d}) \]

of Lemma 22.31.2.

**Proof.**
This follows from Derived Categories, Lemma 13.30.1 and the fact that $- \otimes _ A N$ and $\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, -)$ are adjoint by Lemma 22.30.3.
$\square$

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