Generalization of formal monad theory to lax functors

We study lax functors between bicategories as a generalized concept of monads and describe generalized notions and theorems of formal monad theory for lax functors. Our first approach is to use the 2-monad whose lax algebras are lax functors. We define lax doctrinal adjunctions for a 2-monad $T$ on a 2-category $\mathcal{K}$, and we show that if $\mathcal{K}$ admits and $T$ preserves certain codescent objects, the 2-category $\mathrm{Lax}\text{-}{T}\text{-}\mathrm{Alg}_{c}$ of lax algebras and colax morphisms can coreflectively be embedded in the 2-category of lax doctrinal adjunctions. This coreflective embedding generalizes the relation between monads and adjunctions. Our second approach is to see a distributive law for monads as a 2-functor from a lax Gray tensor product, and we show a generalized form of Beck's characterization of distributive laws.