In this note by Tom Leinster the Banach space $\mathrm{L}^1[0,1]$ is recovered by "abstract nonsense" as the initial object of a certain category of (decorated) Banach spaces. So a function space, that would habitually be defined through the machinery of Lebesgue measure and integration, is uniquely described (up to isomorphism) in terms of abstract functional analysis and a bit of category theory.

I would be curious to see more results, ideally in diverse areas of mathematics, in the spirit of the above one, in which a *familiar* and *important* "concrete" mathematical object is recovered by a universal property (in the technical categorical sense) or -more generally- by a characterizing property that is abstract and general or doesn't delve into the "concrete" habitual definition of that object.

Community wiki, so put one item per answer please.