I realize this question is not research level, but I think this might be just above the abilities of the folks at Mathematics stack. I want a general prescription, or as many as possible, for constructing a monoidal category like FD-Hilb or Hilb given only a monad. I was thinking of using the endo-functor composition as the monoidal product but didn't really get anywhere. We know that the monad axioms are much like the monoidal axioms, though a category like FD-Hilb has many more axioms than can be identified one to one with the monad axioms. Has anyone seen attempts at constructing a basic symmetric monoidal categories from some monad? In particular, I want the monad to be an adjunction of forgetful and free functors, I don't care what the base categories are. Moreover, I want a general prescription: given any free-forgetful adjoint here's how to construct a monoidal category. Help will be most appreciated.