The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of a large class of operadic categories. In particular, we will prove that operads, in the generalized sense of Batanin-Markl, governing important operad- and/or PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diver
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