Question about all the homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}$

If you are working in the category of unitary associative rings, the morphisms $\varphi : R\to S$ must satisfy $\varphi (1)=1$. In this category the ring $\mathbb{Z}$ is an initial object, that is, for any ring $R$ there is exactly one morphism (i.e. ring homomorphism) $\chi : \mathbb{Z}\to R$ (which defines the characteristic of the ring $R$). In particular there is exactly one ring homomorphism $\mathbb{Z}\to\mathbb{Z}$, which is the identity map.

I assume you are talking about Fraleigh's book. If so, he does not require that a ring homomorphism maps the multiplicative identity to itself. Follow his hint by concentrating on the possible values for $f(1)$. If $f$ is a (group) homomorphism for the group $(\mathbb{Z},+)$ and $f(1)=a$, then $f$ will reduce to multiplication by $a$. For what values of $a$ will you get a ring homomorphism? You will need to have $(mn)a=(ma)(na)$ for all pairs $(m,n)$ of integers. What can you conclude about the value of $a$? You still won't have a lot of homomorphisms.