Remark 22.13.6. Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra. Let $M$ be a right differential graded $A$-module. Let $N^\bullet$ be a complex of $R$-modules. The constructions above produce a left differential graded $A$-module $\mathop{\mathrm{Hom}}\nolimits (M, N^\bullet )$ and then a right differential graded $A$-module $\mathop{\mathrm{Hom}}\nolimits (\mathop{\mathrm{Hom}}\nolimits (M, N^\bullet ), N^\bullet )$. We claim there is an evaluation map

$ev : M \longrightarrow \mathop{\mathrm{Hom}}\nolimits (\mathop{\mathrm{Hom}}\nolimits (M, N^\bullet ), N^\bullet )$

in the category of right differential graded $A$-modules. To define it, by Lemma 22.13.2 it suffices to construct an $A$-bilinear pairing

$M \times \mathop{\mathrm{Hom}}\nolimits (M, N^\bullet ) \longrightarrow N^\bullet$

compatible with grading and differentials. For this we take

$(x, f) \longmapsto (-1)^{\deg (x)\deg (f)}f(x)$

We leave it to the reader to verify this is compatible with grading, differentials, and $A$-bilinear. The map $ev$ on underlying complexes of $R$-modules is More on Algebra, Item (17).

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