I've tried to understand what's going on in Example 1.5 on page 27-28 in Hatcher's notes on spectral sequences. There is one part in the reasoning that I can't understand here. He writes down a table with the $E^2$-page of a spectral sequence looking like (arrows omitted)
$$ \begin{array}{ccccccc} \mathbb{Z}a & 0 & \mathbb{Z}ax_2 & 0 & \mathbb{Z}ax_4 & 0 & \ldots\\ \mathbb{Z}1 & 0 & \mathbb{Z}x_2 & 0 & \mathbb{Z}x_4 & 0 &\ldots \end{array} $$
What I'm trying to figure out is how do we know that $ax_2$ is the generator of $E^{2,1}_2$? Hatcher just writes:
The generators for the $\mathbb{Z}$'s in the upper row are $a$ times the generators in the lower row, because the product $E_2^{0,q}\times E_2^{s,t}\to E_2^{s,t+q}$ is just multiplication of coefficients.
Can someone explain to me what's going on here?