Uniqueness In Proof For Fundamental Group Of $S^1$

Question

Perhaps a dumb question, but in the standard proof that $\pi_1(S^1) \cong \mathbb{Z}$ (such as the one given in Hatcher), we prove that every loop $\gamma$ in $S^1$ is homotopic to some loop in $S^1$ of the form $\omega_n(s) = e^{2\pi i n s}$, for $s \in [0,1]$ and $n \in \mathbb{Z}$. But then we prove that $n \in \mathbb{Z}$ is uniquely determined by $\gamma$ and hence there is a unique such $\omega_n$ to which $\gamma$ is homotopic to. Why do we need to prove uniqueness?

Answer 1

Otherwise the fundamental group could be any homomorphic image of $(\mathbb{Z}, +)$. Consider the function that maps every $\omega_n$ to its equivalence class of loops. This is a surjective homomorphism from the group $\{\omega_n \mid n\in \mathbb N\}\cong (\mathbb{Z}, +)$ to the fundamental group. By showing unicity, you obtain that it is also injective, which yields that the fundamental group be isomorphic to $(\mathbb{Z}, +)$.