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Abstract We continue our study of the monoidal category D (G) begun in 12. At the level of cohomology we transfer the duality functor RHom (-, k) to the derived category of dg-modules D (Hₔ^). In the process we develop a more general and streamlined approach to the anti-involution J from 8. We also verify that the tensor product on D (G) corresponds to an operadic tensor product on the dg-side (cf 5). This uses a result of Schnürer on dg-categories with a model structure.
Schneider et al. (Tue,) studied this question.