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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