In most basic abstract algebra courses, the free group is directly constructed, a process that I find rather unwieldy. An alternate method of characterizing the free group is by means of its universal property: for any function $f:S\to G$, an arbitrary group, there is a function $g:S\to F_{S}$ and a unique homomorphism $\varphi: F_{S}\to G$ such that $f=\varphi g$. Of course, a direct construction of the free group is necessary to show that any group actually satisfies this definition. I was wondering what happened when the definition was dualized. In other words, let $P_{S}$ be the group such that for any function $f:G\to S$, there is a function $g:P_{S}\to S$ and a unique homomorphism $\varphi:G\to P_{S}$ such that $f=g\varphi$. It would seem, in light of Cayley's theorem, that $P_{S}$ is just the set of permutations on $S$, but I'm not sure of this. Does anyone know what $P_{S}$ is?