Koszul duality for operadic categories

The aim of this paper (which is a sequel to Operadic categories as a natural environment for Koszul duality) is to set up the cornerstones of Koszul duality and Koszulity in the context of operads over a large class of operadic categories. In particular, for these operadic categories we will study concrete examples of binary quadratic operads, describe their Koszul duals and prove that they are Koszul. This includes operads (for operadic categories) whose algebras are the most important operad- and PROP-like structures such as the classical operads, their variants such as cyclic or modular operads, and also diverse versions of PROPs such as wheeled properads, dioperads, 12PROPs, and still more exotic objects such as permutads and pre-permutads.