A general limit lifting theorem for 2-dimensional monad theory

We give a definition of weak morphism of T-algebras, for a 2-monad T, with respect to an arbitrary family Ω of 2-cells of the base 2-category. By considering particular choices of Ω, 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 Ω above our theorem has as corollaries the lifting of oplax (resp. σ, which generalizes lax and pseudo, resp. strict) limits to the 2-categories of lax (resp. pseudo, resp. strict) morphisms of T-algebras.