My textbook asks:

Choose a homomorphism $\displaystyle \phi:\mathbb{Z}_6\to\mathbb{Z}_9$ with $\displaystyle a,b\in\{0,1,2,3,4,5\}$. Then $\displaystyle \phi([a+b]_6)=\phi([a]_6)+\phi([b]_6)$. What else can we say about $\displaystyle \phi$? Nothing significant, as far as I can tell.Write down the formulas for all homomorphisms from $\displaystyle \mathbb{Z}_6$ into $\displaystyle \mathbb{Z}_9$.

But my textbook has another story. According to it, there are only

First of all, I am not completely certain I understand the notation, here. Is $\displaystyle \phi_n(\alpha)$ simply another way of writing $\displaystyle [\phi(\alpha)]^n$? If so, then by algebra of integers modulo, isn't $\displaystyle \phi_n([x]_6)=\phi([nx]_6)$ true for all $\displaystyle n$? If not, then what does the notation mean?3 different homomorphisms, given by the formulas $\displaystyle \phi_3([x]_6)=[3x]_9$, $\displaystyle \phi_6([x]_6)=[6x]_9$, or $\displaystyle \phi_0([x]_6)=[0]_9$, defined for all $\displaystyle [x]_6\in\mathbb{Z}_6$.

As you can see, I'm a little bit befuddled by this. I would welcome any assistance.

Thanks!