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We establish equalities between cochain and chain type levels of maps bymaking use of exact functors which connect appropriate derived and coderivedcategories. Relevant conditions for levels of maps to be finite are extractedfrom the equalities which we call duality on the levels. Moreover, we give alower bound of the cochain type level of the diagonal map on the classifyingspace of a Lie group by considering the ghostness of a shriek map which appearsin derived string topology. A variant of Koszul duality for a differentialgraded algebra is also discussed.

The ghost length and duality on the chain and cochain type levels