powers of Baire space are homeomorphic to Baire space

I have to show that $N^{\mathbb{N}}$ is homeomorphic to $N$, were N stands for Baire space.

The suggested homeomorphism is this:
$\alpha=(f_0, f_1, \ldots) \in N^{\mathbb{N}}$,
$\phi(\alpha)=(f_0(0), f_0(1), f_1(0), \ldots)$

First of all, I don't understand the definition of that homeomorpism, that is, I don't know what comes next after $(f_0(0), f_0(1), f_1(0), \ldots)$
Second of all, I have problems understanding continuity of functions on N, could someone help proving that the above mapping is indeed homeomorphism?

Thank you for every hint.