This question (more specifically one of the comments) prompted me to ask this question: Can someone give me an example of a mathematical object that one can characterize both via axioms and universal constructions ?
(Please note, that I yet don't have any background in category theory whatsoever, so the only place I came in contact with universal constructions was during an intro course in abstract algebra. So I know how to characterize the field of fractions for example with a universal construction, but generally know very little about universal contructions - I only have a vague intuition about them, which I described more or less in the question I linked)