I'm interested in situations where universal objects come with more structure than their definitions suggest. A classic case of this is where the free abelian group on one element has a ring structure. Proving this is a straightforward exercise using the free-underlying adjunction. So I'd like to know of other cases of this phenomenon, and if possible an explanation as to why the extra structure comes about.

Plenty of examples are given in Hazewinkel's paper Niceness Theorems, but how about very familiar examples such as the rationals? Does, say, the characterisation of $\langle \mathbb{Q}, \gt \rangle$ as the Fraïssé limit of the category of finite linearly ordered sets and order preserving injections tell us why it should support a compatible group, ring and even field structure? Do characterisations of the reals relate to each other?

My question is not completely unrelated to Theorems for nothing (and the proofs for free), as shown by the example given there of subgroups of free groups being free, which also occurs in Hazewinkel's paper.