I was asked to find all homomorphisms $\displaystyle \Phi :\mathbb{Z}\to\mathbb{Z}$

I'm fairly convinced in my own mind that all of them are $\displaystyle \Phi (x)=cx \ \forall \ c \in \mathbb{Z}$

I can show that these are homomorphisms by claiming that $\displaystyle \Phi$ is defined as above and:

1. $\displaystyle \Phi (a+b)=\Phi(a)+\Phi(b)$ (definition of homomorphism)

2. $\displaystyle c(a+b)=c(a)+c(b)$ (1, my def. of $\displaystyle \Phi$)

3. $\displaystyle ca+cb=ca+cb$ (2, distributive prop.)

So, how do I show now that there are no other homomorphisms?