First I will state several theorems (old and new) which deal with operads, their generalizations, and their algebras.  Second, I will explain how each of these theorems can be rephrased as an intertwining statement; by this I roughly mean they describe the image of a composition DfD where f is a functor and D means dualize.  Lastly, I will explain how intertwining statements arise as a formal consequence of categorical structure which is present in these examples.  Our conclusion will thus be new proofs of these old and new theorems.