Abstract: 2-monads are monads in the 2-category 2-CAT of large 2-categories. Strict algebras for a suitably chosen 2-monad T on a 2-category K can describe all sorts of algebraic structures: monoidal categories, symmetric monoidal categories, categories with a monad, categories with chosen limits, and so forth. In these examples the appropriate morphisms of algebras are variously pseudo, lax, or colax, not strict. We begin by considering adjunctions in the 2-category for a monad on CAT of strict T-algebras, lax morphisms, and the appropriate 2-cells. An easy diagram chase shows that, a right adjoint in CAT has a lax T-morphism structure if and only if its left adjoint has a colax structure. It follows that left adjoints in the 2-category described above are necessarily pseudo, not lax. One consequence is the familiar fact that left adjoints preserve colimits.
We conclude with a sequence of "coherence" theorems for algebras and morphisms with varying degrees of weakness. With appropriate hypotheses on the 2-category K and the 2-monad T, the inclusions of the 2-category of T-algebras and strict morphisms into, respectively, the 2-categories of T-algebras and pseudo morphisms, T-algebras and lax morphisms, and pseudo T-algebras and pseudo morphisms all have left adjoints that can be constructed explicitly. The unit of the last adjunction is an equivalence A ~> A' between a pseudo algebra A and a strict algebra A'. One consequence of this result is that every (unbiased) monoidal category is equivalent to a (particular) strict one.