In Semigroup Theory, Green's relations are everywhere. Their equivalence classes, for instance, on a given semigroup $S$ can tell one a lot about the structure of $S$.
There is some trivial sense in which a monoid can be viewed a category. Some people even argue that (at least small) categories are generalised monoids.
In light of the above, then, it seems natural to ask how to describe Green's relations using Category Theory, even if it is just for the category $\textbf{Monoids}$ and not $\textbf{Semigroups}$ or beyond.
If I had the time or the ability, I'd look into Lawvere theories a little deeper in my attempt to answer this question. I'm not entirely sure why. Perhaps it's because Universal Algebra is more visible in the structural approaches to Semigroup Theory I've seen.
I'm just curious :)
There is no need of category theory to define Green relations on categories. Let $C$ be a small category and let $0$ be a new element. Then $C \cup \{0\}$ is semigroup for the operation $*$ defined as follows: $$ x*y = \begin{cases} xy & \text{if $xy$ is defined in $C$}\\ 0 & \text{otherwise} \end{cases} $$ You can now just define the Green relations on this semigroup.