  Jean-Louis Loday has defined generalised bialgebras and proved structure
theorems in this setting which can be seen as general forms of the
Poincar\'e-Birkhoff-Witt and the Cartier-Milnor-Moore theorems. It was observed
by the present authors that parts of the theory of generalised bialgebras are
special cases of results on entwined monads and comonads and the corresponding
mixed bimodules. In this article the Rigidity Theorem of Loday is extended to
this more general categorical framework.
