Is there some sort of monad whose algebras are monads? How about if we are internal to a bicategory B? Are internal monads in B monadic? Certainly not always, as otherwise free T-multicategories a la Leinster would always exist. What is known?



Are monads monadic?
In the book "Toposes, Triples and Theories", Barr and Wells study the question when a *particular* endofunctor admits a free monad. This is the case if the underlying category is complete and cocomplete and if the endofunctor preserves filtered colimits (we say that such a functor is *finitary*).


The question remains whether the resulting adjunction between monads on $C$ and endofunctors on $C$ is monadic. If $C$ is [locally finitely presentable](http://ncatlab.org/nlab/show/locally+presentable+category) (lfp), this is true: Steve Lack showed in


"On the monadicity of finitary monads", Journal of Pure and Applied Algebra, Volume 140, Number 1, 21 July 1999 , pp. 65-73(9) (available [here](http://www.maths.usyd.edu.au/res/Catecomb/Lack/1997-29.html))


that the forgetful functor $\mathrm{Mnd}\_f(C) \rightarrow \mathrm{End}\_f(C)$ from finitary monads on a lfp category $C$ to finitary endofunctors is monadic. Note that both these categories are again lfp categories, and we don't need universes to make sense of them, essentially because a finitary endofunctor is determined by what it does on a set of objects.


The result remains true if we consider categories and functors enriched in a complete and cocomplete symmetric monoidal closed category $V$ which is lfp as a closed category.


