It is a standard consequence of the Brown Representability Theorem for $\operatorname{Ho}(\operatorname{Top}_*)$ that the category of generalized cohomology theories for spaces (pointed CW complexes, more specifically) is equivalent to the stable homotopy category $\operatorname{SHC}$ (defined as the homotopy category of the stable model structure on sequential spectra) via representability. On the other hand, we can define a cohomology theory for spectra (as in Barnes and Roitzheim's *Foundations of Stable Homotopy Theory*) to be a contravariant functor $E^*:\operatorname{SHC}^{op}\to\operatorname{Ab}_*$ such that

- Each exact triangle in $\operatorname{SHC}$ gives rise to a long exact sequence in $\operatorname{Ab}$
- $E^n$ preserves products for each $n$ (that is, it sends wedge sums to products of abelian groups)
- $E^*$ preserves suspension up to a specified natural isomorphism $E^{n+1}(\Sigma X)\cong E^n(X)$.

Then it is easy to check that $[,E]$ is always a cohomology theory for spectra. But is every cohomology theory for spectra thus represented? Given such a cohomology theory $E^*$, we obtain an associated cohomology theory for spaces by restricting to suspension spectra. Thus, by the result for spaces mentioned above, the question becomes whether a cohomology theory for spectra is determined by its restriction to spaces.

This is certainly true if we require $E^*$ to preserve sequential homotopy colimits, because any spectrum is weakly equivalent to a CW spectrum with basepoint at its unique $0$-cell, which is a sequential homotopy colimit of homotopy cofibers of coproducts of shifted sphere spectra. But without this requirement, does the result still hold? If so, why? If not, is there a standard counterexample?