I'm having trouble understanding the last part of Hatcher's proof of Hurewicz' theorem. (It's on page 367, thm. 4.32).
We want to show, that a cellular boundary map: $d:H_{n+1}(X^{n+1},X^n) \rightarrow H_n(X^n,X^{n-1})$ is a map $\oplus_\beta \mathbb{Z} \rightarrow \oplus_\alpha \mathbb{Z}$, where $X$ only has cells in dimensions $n$ and $n+1$ and is given by $(\bigvee_\alpha S_\alpha ^n )\bigcup_\beta e^{n+1}_\beta.$
The argument goes like this:
"$d$ [is a map $\oplus_\beta \mathbb{Z} \rightarrow \oplus_\alpha \mathbb{Z}$] since for each cell $e_\beta ^{n+1}$, the coefficients of $de_\beta ^{n+1}$ are the degrees of the compositions $q_\alpha \phi_\beta$ where $q_\alpha$ collapses all n-cells except $e_\alpha ^n$ to a point, and the isomorphism $\pi_n (S^n) \approx \mathbb{Z}$ in corollary 4.25 us given by degree."
Any help or ideas would be welcome :)