Can one tell whether an (abstract) monoid $M$ is (isomorphic to) the monoid of endomorphisms of a structure $X$?
Or is there a representation theorem saying that for every monoid $M$ there is a structure $X$ such that $M$ is the monoid of endomorphisms of $X$? If not so: what is a simple counter-example?
How can the monoids which are monoids of endomorphisms be characterized?
Cayley's theorem for monoids asserts that $M$ is isomorphic to the monoid of endomorphisms of $M$ as a right $M$-set. This argument can be internalized to various categories; for example, a ring $R$ is isomorphic to the ring of endomorphisms of $R$ as a right $R$-module.