Are there references for presenting common categories up-to equivalence using universal properties? I'm familiar with the Elementary Theory of the Category of Sets (ETCS), which provides axioms (mostly universal properties) which determine the category of sets and functions up-to equivalence. Similar projects exist for certain categories of topological spaces, but I haven't seen much work on algebraic categories.
For example, is there a collection of universal properties characterizing the category of monoids in set up-to equivalence? If this were the case, then one could use an unambiguous internal language for the category of monoids.