Let $\aleph^{op}_{0}$ be a skeleton of the category of finite Sets and all functions between them.

A Lawvere theory consists of a small category, $L$ with (necessarily strict associative) finite products and a strict finite product-preserving identity on objects functor $I: \aleph^{op}_{0} \rightarrow L$. A map of Lawvere theories $f: L \rightarrow L'$ is a (necessary strict product preserving) functor that commutes with the functors $I$ and $I'$.

We can generate a monad from a Lawvere Theory in the following way:

The Monad $T_L$ may be described by the following colimit: $$T_LX = \int^{n \in \aleph_0} L(n,1)\times X^n $$

Can we find a similar definition of a category that is a theory in the same sense of a Lawvere theory, but that generates a (Co)Monad on Set, ie both a monad and comonad? Is there a way to define the (co)unit and (co)multiplication?

There is a bit of work in this direction by Behrisch, Kerkhoff,and Power.

As a concrete example, we have the List Monad, which has a Lawvere theory, that of Monoids. There is a bimonad that is an extension of the List Monad, namely the "Tails" comonad. We see it here. What would be the theory of this bimonad? Can we phrase it like a Lawvere theory?