  Generalizing the approach to pseudo monoidal DG-categories as certain colored
non-symmetric DG-operads, we introduce a certain relaxed notion of a category
enriched in DG-categories. We construct model structures on the category of
colored non-symmetric DG-operads and on the category of DGCat-enriched
categories with a fixed set of objects. This allows us to talk about strong
homotopy maps in both settings. We discuss the notion of a strong homotopy
monad in a DG-category and a notion of strong homotopy adjunction data for two
DG-functors.
