I'm currently reading some notes about the James spectral sequence (here) and there is a passage which is bothering me (page 749 bottom):

$$ colim_n h_{p+q}(T(\xi_n))\cong h_{p+q}(M\xi)$$

where $M\xi$ is a Thom spectrum whose component space are $\{ T(\xi_n), s_n\colon S^1 \wedge T(\xi_n) \to T(\xi_{n+1})\}$ and $\xi_n$ is a $n$-dimensional orientable vector bundle over $B_n$. (The precise definition of all objects here is unnecessary according to me).

I'm unable to show the highlighted congruence, since my definition of homology of a spectrum is $$h_k(M\xi)= colim_n h_{k+n}(T(\xi_n))$$ and I can't get rid of the dependancy of $h_{k+n}$ to the index $n$ as suggested by the formula above. Am I Missing something?


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