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](https://ncatlab.org/nlab/show/adjunction#general) says:



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> 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.
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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).



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



"Equivalence" is to "group" as "adjoint" is to ....?
If everything has both a left and a right adjoint, then you're talking about a rigid monoidal category. (Here I've done the usual dimension shift where a 2-category with one object is the same as a monoidal category.)


If you only want adjoints on one side it's a bit more awkward because that's not left rigid (since the left adjoint will only have a right adjoint) nor right rigid. But for the single functor case it'd be a monoidal category generated by a right rigid object.


