categories: Kleisli and colimits

[Tom Leinster <T.Leinster@dpmms.cam.ac.uk>]

Does anyone know if the Kleisli construction (sending a monad to its Kleisli
category) behaves in any decent way with respect to colimits?  E.g. does it
in any sense preserve or reflect them?

The actual situation that I have is a fixed category C, and a certain
coequalizer diagram in the category of monads on C.  The resulting fork in
Cat is also a coequalizer, and the proofs that both diagrams are coequalizers
have some ingredients in common, but I can't at present see how to deduce one
from the other.  

Thanks,

Tom Leinster