Im trying to show that:
For every homomorphism $\varphi:\Pi_1(S^1,1)\rightarrow\Pi_1(S^1,1)$ there is a pointed map $f:(S^1,1)\rightarrow{(S^1,1)}$ so that $\varphi=f_*$. Namely, $f$ induces $\varphi$.
Im not too sure were to start with this. We know from the homomorphism property that $\varphi([f*g])=\varphi([f]\bullet[g]])=\varphi([f])\bullet\varphi([g])$.
A group homomorphism $\varphi:\Bbb Z\to\Bbb Z$ is determined by its value $n=\varphi(1)$, and must be of the form $\varphi=x\mapsto nx$.
Now consider $S^1$ as the set of unit complex numbers, with $1$ as the distinguished point, and verify that $$f=x\mapsto x^n$$ induces $\varphi$ on the homotopy group.