Lemma 22.13.2. Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded $R$-algebra. Let $M'$ be a right differential graded $A$-module and let $M$ be a left differential graded $A$-module. Let $N^\bullet$ be a complex of $R$-modules. Then we have

$\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}_{(A, d)}}(M', \mathop{\mathrm{Hom}}\nolimits (M, N^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}(R)}(M' \otimes _ A M, N^\bullet )$

where $M \otimes _ A M$ is viewed as a complex of $R$-modules as in Section 22.12.

Proof. Let us show that both sides correspond to graded $A$-bilinear maps

$M' \times M \longrightarrow N^\bullet$

compatible with differentials. We have seen this is true for the right hand side in Section 22.12. Given an element $g$ of the left hand side, the equality of More on Algebra, Lemma 15.71.1 determines a map of complexes of $R$-modules $g' : \text{Tot}(M' \otimes _ R M) \to N^\bullet$. In other words, we obtain a graded $R$-bilinear map $g'' : M' \times M \to N^\bullet$ compatible with differentials. The $A$-linearity of $g$ translates immediately into $A$-bilinarity of $g''$. $\square$

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