Construction of 2-category of monoidal categories and (lax) monoidal functors as strict algebra category of a 2-monad

Question

As motivation for 2-monads, I would like to understand an explicit construction of the 2-monad $T$ of which derived 2-category $T-\operatorname{Alg}_l$ of algebras as described in Lack's 2-categories companion produces the 2-category of monoidal categories, lax monoidal functors and lax natural transformations. I would like $T$ very explicitly specified in contrast to e.g. this which uses as starting point "free monoid construction" which makes sense to me only as functor $\operatorname{Set} \to \operatorname{Mon} \to \operatorname{Cat}$ instead as what Lack says being $\operatorname{Cat} \to \operatorname{Cat}$.