Let $\xi_n$ be an orientable $n$-dimensional vector bundle over a pointed space $B_n$. We can consider the relative Serre Spectral Sequence $$ H_p(B_n; h_q(D(\xi_n|\ast),S(\xi_n|\ast))\Rightarrow h_{p+q}(D(\xi_n),S(\xi_n)) $$ which can be rewritten as $$ H_p(B_n; \tilde{h}_q(S^n))\Rightarrow \tilde{h}_{p+q}(M\xi_n) $$ where $M\xi_n$ is the Thom Space of the vector bundle $\xi_n$. Via the suspension isomorphism we have equivalently $$ H_p(B_n; \tilde{h}_{q-n}(S^0))\Rightarrow \tilde{h}_{p+q}(M\xi_n) $$ using the Thom-Isomorphism $H_p(M\xi_n)\cong H_{p-n}(B_n)$ we get $$ H_{p+n}(M\xi_n; \tilde{h}_{q-n}(S^0))\Rightarrow \tilde{h}_{p+q}(M\xi_n)$$ which *seems* the Atiyah-Hirzebruch Spectral sequence for $M\xi_n$ in the sense that I'm unable to prove that *under these transformations* the differentials coincide with the one that comes from the AHSS, in other words that what I got is really the AHSS. How can I show that this is indeed the case?

Since these transformations are not given by a geometrical map, I'm unable to boil everything down to a simply naturality argument.

I'm interested in this fact because I'm reading this paper about the construction of the James Spectral Sequence where this fact is crucial for the identification of the second differential of this spectral sequence (Prop 1) and seems to be trivial.