Wondering for the ncatlab I've encountered the following pages: one about lax-morphisms and the other about lax-algebras for $2$-monads.
For what I could get it seems that lax algebras can be characterized as lax morphisms in a category of strict algebras for a strict $2$-monad having as algebras the strict monads.
Could anyone write down more explicitly this construction or link any reference where such characterization is spelled out?