  We give a definition of weak morphism of $T$-algebras, for a $2$-monad $T$,
with respect to an arbitrary family $\Omega$ of $2$-cells of the base
$2$-category. By considering particular choices of $\Omega$, we recover the
concepts of lax, pseudo and strict morphisms of $T$-algebras. We give a general
notion of weak limit, and define what it means for such a limit to be
compatible with another family of $2$-cells. These concepts allow us to prove a
limit lifting theorem which unifies and generalizes three different previously
known results of $2$-dimensional monad theory. Explicitly, by considering the
three choices of $\Omega$ above our theorem has as corollaries the lifting of
oplax (resp. $\sigma$, which generalizes lax and pseudo, resp. strict) limits
to the $2$-categories of lax (resp. pseudo, resp. strict) morphisms of
$T$-algebras.
