Completing proof of theorem 3.2 in Yiannis Loizides' The Atiyah-Hirzebruch Spectral Sequence.

Question

Let $X$ be a CW-complex, and denote its $k$-skeleton by $X_k$.

Let $h^*$ be a cohomology theory, and consider its AHSS for $X$: call $E_n^{p,q}$ the groups in the $n$-th page, and call $d_n^{p,q}$ the $p,q$-component of the differential map $d_n:E_n\to E_n$, where $E_n=\bigoplus_{p,q}E^{p,q}_n$.

I am trying to follow the proof in Yiannis Loizides' notes that this fact holds, where $h^q:=h^q(S^0)$: $$E^{p,q}_2= H^p(X,h^q).$$

From what I see, the notes prove that $E_1^{p,q}=H^{p+q}(X_{p+1}/X_p,h^q)$, and that $d_1^{p-1,q}:H^{p+q-1}(X_{p}/X_{p-1},h^q)\to H^{p+q}(X_{p+1}/X_p,h^q)$ is a known map, the coboundary map.

The rest of the proof is left implicit; I think it should use that by definition $E_2^{p,q}=ker(d_1^{p,q})/im(d_1^{p-1,q})$, and this quotient is $H^p(X,h^q)$.

However I don't understand (1) in what sense the components of $d_1$ are coboundary maps (they don't look like the coboundary maps in the long exact sequence), and (2) if the final part of the proof is actually like I expect; if yes, would you give me a bit of context on how to identify that quotient of a subgroup in $E_1^{p,q}$ with $H^p(X,h^q)$.

Edit after comments. Set $m:=p+q$. Now I see now that:

• $d_1^{p-1,q}:H^{m-1}(X_{p}/X_{p-1})\to H^{m}(X_{p+1}/X_p)$ is a coboundary map in the long exact sequence associated to $(X_{p+1}, X_p, X_{p-1})$;
• $d_1^{p,q}:H^{m}(X_{p+1}/X_{p})\to H^{m+1}(X_{p+2}/X_{p+1})$ is a coboundary map in the long exact sequence associated to $(X_{p+2}, X_{p+1}, X_{p})$.

So maybe I can define better my issues:

1. The isomorphism between cellular and singular cohomology says that $H^n(X_n,X_{n-1})=H^n(X)$, but this kind of indexing does not necessarily occur in the cellular cohomology groups above; it occurs if $m=p-1$ for example.
2. As a consequence of the previous issue, I don't see how to traduce these consecutive coboundary maps $$H^{m-1}(X_{p}/X_{p-1})\xrightarrow{d_1^{p-1,q}} H^{m}(X_{p+1}/X_p)\xrightarrow{d_1^{p,q}} H^{m+1}(X_{p+2}/X_{p+1})$$ into maps of singular cohomology groups, in order to understand why $ker(d_1^{p,q})/im(d_1^{p-1,q})=H^p(X)$.