Let $G$ be a group such that there is a homomorphism $f\colon\mathbb{Z}\to G$ and such that for every group $H$ and every homomorphism $\phi\colon\mathbb{Z}\to H$ there is exactly one homomorphism $\xi\colon G\to H$ such that $\xi\circ f=\phi$.
Show that $G$ is isomorphic to $\mathbb{Z}$.
I think that this is rather very easy though I can't see it ...
Thanks in advance