Combining monads

Is the pullback of a monadic functor along a monadic functor necessarily monadic? Is the diagonal of the pullback square monadic? Does this work if your restrict yourself to, say, finitary monadic functors? (E.g. it works for finitary monads on Set: the theory of sets with both ring and lattice structure (not interacting in any particular way) comes from a monad.) Thanks, Tom Leinster