The collection of all self-equivalences of a category $C$ constitutes a $2$-group, which is a categorification of the notion of a group. My question is about what happens when one replaces equivalences by general adjoint functors.

The n-lab page on adjunctions says:

a morphism in an adjunction need not be invertible, but it has in some sense a left inverse from below and a right inverse from above.

I would like to know if this can be formalised in an interesting way.

In order to ask a specific question, I want to take a single adjoint pair of functors $(L,R):C\to C$, and let $M$ be the monoid of endofunctors of $C$ generated by $L$ and $R$ (together with some suitably chosen family of natural transformations).

Q.What kind of ($2$-)algebraic structure is $M$?

I apologise that this is a somewhat vague question. An ideal answer would have the form "$M$ is a (lax/colax/...) $2$-blah", where "blah" is an interesting algebraic structure that arises in other contexts not a priori having anything to do with adjoint functors. If it makes more sense to change the setting a little---e.g., to consider instead of $M$ the collection of all functors on $C$ admitting a two-sided adjoint---then please go ahead and do so.