It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way he proved that operads cannot be identified with monads. Now I'm wondering: is it true that every cartesian monad arises from an operad, i.e. every such a monad is the monad associated to an operad?


I little specification, what I mean here by operad is a generalized operad i.e. a $T$-operad for some cartesian monad $(T,\mu,\eta)$ in a cartesian category $\mathcal C$.



Monad arising from operad
The answer is no, and it is explained in Appendix C of Leinster's book *Higher Operads, Higher Categories*.


Briefly, a 'plain operad', for Leinster, is a non-symmetric operad in Set; a T-operad for T the free-monoid monad. Leinster shows that if T is a cartesian monad then a T-operad is the same thing as a cartesian monad S equipped with a cartesian transformation $S \to T$. So a plain operad is the same thing as a cartesian monad on Set that is 'augmented over' the free-monoid monad. But not every cartesian monad on Set is so augmented.


