TITLE: Spectra as functors from Spaces to Spaces
QUESTION [9 upvotes]: I will use the notation of this question. So, if $X$ is a (nice) topological space and $G$ is an abelian group, we can form its $G$-linearization $G[X]$. In McCord's article, this was denoted $B(G,X)$.
In that question it is mentioned that $\pi_*(G[X])=\tilde{H}_*(X;G)$. But more is true: the functor from Spaces to Graded Abelian Groups that maps a space $X$ to $\pi_*(G[X])$ is a homology theory, isomorphic to singular homology with coefficients in $G$. This says that the isomorphism commutes with the boundary maps in long exact sequences. And we have a good grip on what the boundary for $\pi_*(G[-])$ is, actually: if $A\to X$ is a cofibration, then McCord proves that
$$G[A]\to G[X]\to G[X/A]$$
is a fibration (actually, a principal bundle). The boundary map in homology for $A\to X$ corresponds to the boundary map in homotopy for this fibration.
With this set up, we have that $H_*(-;G)$ is naturally isomorphic to the composition $\pi_* \circ G[-]$.
If I understand correctly what I've heard, it is actually true that for any spectrum $E$ we have that $E_*$ (the associated homology functor from Spaces to Graded Abelian Groups) decomposes as a composition $\pi_* \circ E[-]$, where $E[-]$ is some functor from spaces to spaces which maps cofibrations to fibrations.

Question 1: what further hypotheses do we need on these functors to make the correspondence with the category of homology theories be a 1-1 correspondence? We certainly need $E[-]$ to map a point to a point (or to something contractible). Is this enough?

So, for $E=HG$, the Eilenberg-Mac Lane spectrum of $G$, we have that $E[-]=G[-]$: for $H\mathbb Z$, for example, $E$ is the infinite symmetric product functor.
For $E$ the sphere spectrum, we have $E[-]=Q$.

Question 2: what can be said about other spectra? Is there a clean description on how to get $E[-]$ from $E$ in general? Or maybe in other well-known cases, $KO, KU, MO, MU, K(n)$, etc...

Subsidiary question for the comments: any further references to this point of view are welcome.

REPLY [4 votes]: As a starting point you might well move backwards in time to 1960:
MR0115163
Whitehead, George W.
Homology theories and duality.
Proc. Nat. Acad. Sci. U.S.A. 46 1960 554–556. 
The details of the results announced there were written up in 
Whitehead, George W.
Generalized homology theories.
Trans. Amer. Math. Soc. 102 1962 227–283. 
The representability of homology theories in terms of spectra dates from then. I encourage you to read the Math Review of the first by Peter Hilton.  The representing spectrum is not unique, but I think the lim^1 error term measuring non-uniqueness was understood when I was a graduate student, or very soon thereafter. (I got my PhD in 1964). A clean exposition is in Section 1 of McClure's Chapter VII of $H_{\infty} ring spectra and their applications, published in 1986.