If is a 2-monad on a 2-category , then in addition to strict (if and are strict) -algebras, which satisfy their laws strictly, and pseudo -algebras, which satisfy laws up to isomorphism, one can consider also lax and colax algebras, which satisfy laws only up to a transformation in one direction or the other.
If is a 2-monad on , a lax -algebra in consists of an object , a morphism , and (not necessarily invertible) 2-cells
satisfying suitable axioms. (Here is the multiplication of and is the unit.)
Of course, in a colax -algebra the transformations go the other way. The official way to remember which is lax and which is colax is that a lax -algebra structure on is a lax M-morphism , where is the 2-monad on the 2-category of (some) endofunctors of whose algebras are 2-monads, and is the codensity monad of , i.e. the right Kan extension of along itself.