  Generalizing a concept of Lipshitz, Ozsv\'ath and Thurs-ton from Bordered
Floer homology, we define $D$-structures on algebras of unital operads, which
can also be interpreted as a generalization of a seemingly unrelated concept of
Getzler and Jones. This construction gives rise to an equivalence of derived
categories, which can be thought of as a unital version of Koszul duality using
non-unital Quillen homology. We also discuss a multi-sorted version of the
construction, which provides a framework for unifying the known algebraic
contexts of Koszul duality.
