What homological functors $\mathbf{Sp}\rightarrow\mathbf{Ab}$ arise in the form $H_0\circ T$, for $T:\mathbf{Sp}\rightarrow D(\mathbf{Ab})$ a triangulated functor, and dually for cohomological functors $\mathbf{Sp}^{\mathrm{op}}\rightarrow\mathbf{Ab}$? What if I replace $\mathbf{Sp}$ with a derived category? Here, $D(\mathbf{Ab})$ is the derived category of unbounded complexes of Abelian groups, $\mathbf{Sp}$ the triangulated category of spectra, and $H_0:D(\mathbf{Ab})\rightarrow\mathbf{Ab}$ the usual homology functor.
Phrased less formally, what (co)homology theories of spectra can be presented by a natural complex(?) of $D(\mathbf{Ab})$? For context, I am trying to understand triangulated/derived category formalism, and this seems an important baseline to establish.
Ordinary homology with $A$ coefficients can be presented this way by letting $TX$ be the limit $\varinjlim C_\bullet\left(X_n;A\right)$ of the cellular complexes of the spaces $X_n$ comprising the spectrum $X$. On the other hand, a proposition on page 35 of a set of notes by Krause seems to imply one cannot factor $\pi^S_0$ through $H_0$. My question seems closely related to this one, based on which it seems plausible to guess the answer is that ordinary (co)homology is the only answer up to translation and direct sum (product?).
As a related-but-not-quite-the-same question, I was trying the following approach before the above discouraged me from working out more details. Consider a cohomology theory $E^*$ of pointed CW complexes represented by an $\Omega$-spectrum $\left(E_n\right)$. One has a sequence of pointed spaces
$$ \cdots\rightarrow \mathcal{P}E_{n-1} \rightarrow \mathcal{P}E_n \rightarrow \mathcal{P}E_{n+1} \rightarrow \cdots $$
given by stringing together the path space fibrations $\Omega E_n\rightarrow\mathcal{P}E_n\rightarrow E_n$ of the spaces $E_n$. It seems with a little care one should be able to describe a cochain complex from this sort of thing, closely-related enough to path space fibrations to have meaningful cohomology. (One has to be more careful than "take homotopy classes of maps" since $\mathcal{P}E_n$ is contractible.) Does this sort of thing lead anywhere? If so, the only reasonable places I could see this ending up are a tautology or the vicinity of an Atiyah-Hirzebruch spectral sequence.
I am a number theory student, so I apologize if some technical details regarding algebraic topology are a little vague or not quite correct.
