On page 145 of J.F. Adams' "Stable Homotopy and Generalised Homology", there is a proposition:
Let $E$ be the suspension spectrum of a finite CW-complex $K$, and $F$ and spectrum (of topological spaces). Then $[E,F]=\mathrm{colim_n}[\Sigma^nK,F_n]$. The proof is not difficult, and it proceeds as follows:
Given two spectra maps which agree in the colimit, say $f$ and $g$, they agree on some finite level, $[\Sigma^pK,F]$. Thus, the homotopy at that level can be suspended to create a homotopy of cofinite subspectra, which is exactly what we need to have a homotopy class of maps in $[E,F]$. However, I cannot see why he requires that $K$ be a finite CW-complex. It seems that we can find an upper bound of $f$ and $g$ regardless of that constraint, based on the fact that we are using a filtered colimit. Does this make sense, or is there an obvious reason why one might choose a finite $K$?
Thanks!