I'm trying to understand the proof in Hatcher (p. 141) of the cellular boundary formula. Now there's one thing that Hatcher does several times in his book and that I don't understand very well: he states things such as "In terms of the basis for $H_{n - 1}(X^{n - 1}, X^{n - 2})$ corresponding to the cells $e^{n - 1}_{\beta}$, the map $q_{\beta *}$ is the projection of $H_{n - 1}(X^{n - 1}/X^{n - 2})$ onto its $\mathbb{Z}$ summand corresponding to $e^{n - 1}_{\beta}$". Here $X$ is a CW-complex, $e^{n- 1}_{\beta}$ is a $(n - 1)$-cell and $q_{\beta}$ is the map $X^{n -1}/X^{n-2} \to S^{n-1}_{\beta}$ that "collapses the complement of the cell $e_{\beta}^{n-1}$ to a point, the resulting quotient sphere being identified with $S_{\beta}^{n-1} = D_{\beta}^{n - 1}/\partial D_{\beta}^{n - 1}$ via the characteristic map $\Phi_{\beta}$."
Now that seems intuitively plausible and I guess I can prove it by using chains and such things, but that doesn't seem very elegant. What's the right way to understand this fact? It might be very simple but I can't see a way that seems right to me.
Thanks!