The free group $F(S)$ on a set $S$ is a [cogroup][1] in the category of groups since $\hom(F(S),G) \cong G^S$ carries a natural group structure for every group $G$. I have read that these are the only cogroups in the category of groups. This result is attributed to Kan:

> Daniel M. Kan, On monoids and their dual, Bol. Soc. Mat. Mexicana (2) 3 (1958), 52–61. MR 0111035 (22 #1900)
 
<strike>However I have no access to this paper</strike>, and could not find it online either. Perhaps someone knows the paper and can give me a hint how to prove the result? Thanks a lot.

Edit: Tyler's answer explains why the underlying group of any cogroup is free. I accept it because meanwhile I've found Kan's paper.


  [1]: http://ncatlab.org/nlab/show/cogroup