Lemma 22.19.2. Let $(A, \text{d})$ be a differential graded algebra. If $M$ is a left differential graded $A$-module and $N$ is a right differential graded $A$-module, then

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}_{(A, \text{d})}}(N, M^\vee ) & = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}(\mathbf{Z})}(N \otimes _ A M, \mathbf{Q}/\mathbf{Z}) \\ & = \text{DifferentialGradedBilinear}_ A(N \times M, \mathbf{Q}/\mathbf{Z}) \end{align*}

Proof. The first equality is Lemma 22.13.2 and the second equality was shown in Section 22.12. $\square$

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