  We construct a `weak' version EM^w(K) of Lack & Street's 2-category of monads
in a 2-category K, by replacing their compatibility constraint of 1-cells with
the units of monads by an additional condition on the 2-cells. A relation
between monads in EM^w(K) and composite pre-monads in K is discussed. If K
admits Eilenberg-Moore constructions for monads, we define two symmetrical
notions of `weak liftings' for monads in K. If moreover idempotent 2-cells in K
split, we describe both kinds of a weak lifting via an appropriate
pseudo-functor EM^w(K) --> K. Weak entwining structures and partial entwining
structures are shown to realize weak liftings of a comonad for a monad in these
respective senses. Weak bialgebras are characterized as algebras and
coalgebras, such that the corresponding monads weakly lift for the
corresponding comonads and also the comonads weakly lift for the monads.
