I don't really have any experience working with universal maps or adjoint functors (well, neither with categories, really. Only some basic definitions and intuitions). However, I have the feeling that maybe it's not a crazy idea to think of multiplicative properties in topology as topological invariants preserved when formal categorical products (or coproducts, or quotients... ) are done. I guess this topological products, coproducts and quotients have to do with universal maps, that's why I picked that title for my question, but I don't know how to apply this language yet so I'm sorry in advance for any possible misunderstanding.
Is it possible to achieve some statement like "Because of these categorical properties of compactness, compactness is a multiplicative property"?
Although it is a question that I made two years ago, I just realised after discussing it with a collegue that this question is a particular case to something that every mathematician is constantly trying to answer: "do these two things commute?"
That was, at least, what I intended to understand with this question, although I did not know it back then.
In any case, asking for a categorical characterization of a topological property that would let us completely understand its behavior with respect to the usual operators of set theory or algebra is like asking for "a general solution" to Category Theory.
Realising this, at least for me, answers the question.