  A weak entwining structure in a 2-category K consists of a monad t and a
comonad c, together with a 2-cell relating both structures in a way that
generalizes a mixed distributive law.A weak entwining structure can be
characterized as a compatible pair of a monad and a comonad, in 2-categories
generalizing the 2-category of comonads and the 2-category of monads in K,
respectively. This observation is used to define a 2-category Entw^w(K) of weak
entwining structures in K. If the 2-category K admits Eilenberg-Moore
constructions for both monads and comonads and idempotent 2-cells in K split,
then there are pseudo-functors from Entw^w(K) to the 2-category of monads and
to the 2-category of comonads in K, taking a weak entwining structure (t,c) to
a `weak lifting' of t for c and a `weak lifting' of c for t, respectively. The
Eilenberg-Moore objects of the lifted monad and the lifted comonad are shown to
be equivalent. If K is the 2-category of functors induced by bimodules, then
these Eilenberg-Moore objects are isomorphic to the usual category of weak
entwined modules.
