Let H be a group containing an element h with the following property: given a group G and an element g in G, there is a unique homomorphism from H to G mapping h to g. Prove that H is isomorphic to Z (the set of integers).

I know (or I think) that such that defines an isomorphism from H to Z. It's a homomorphism and it is surjective since 1 is a generator of Z. But I'm having a tough time proving that it is injective. Any hints or advice? Thanks.