TITLE: Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$
QUESTION [6 upvotes]: This question relates to Realizing groups as automorphism groups of graphs.
Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?

REPLY [12 votes]: Trying to forestall another question along these lines let me add that every locally presentable category fully embeds into the category of graphs - see Adámek, Rosický
Locally presentable and accessible categories 1994.
In fact, depending on set theories in which we work, every concrete category fully embeds into graphs - see Pultr, Trnková, Combinatorial, algebraic and topological representations of groups, semigroups and categories 1980.
For example: the category of metrizable spaces embeds into Graphs and in some set theories the category of Hausdorff topological spaces also embeds into Graphs.
Monoid is a category with a single object, hence a special case of the first result.