Assume that $\{E^n\}_{n \in \mathbb{Z}}$ is a cohomology theory, that is, a sequence of functors from the (homotopic) category of based CW-complexes to the category of (graded) abelian groups. Then Brown's representability theorem asserts that for any such sequence, there is always a nice sequence of spaces $\{ Z^n \}_{n \in \mathbb{Z}}$, which represents the functors $E^n$. In other words we have a natural equivalence, $E^n \cong [-,Z^n]$.
My question has to do with the "dual" version of the above narrative. Assume that someone starts with a a spectrum $Z=\{Z^n\}_{n \in \mathbb{Z}}$. Then Switzer in his book "Algebraic Topology – Homotopy and Homology", makes the following constrcution: for a given spectrum $Z$, defines a cohomology theory $E$ associated to $Z$, with $E^n(X)=[\Sigma^{\infty}X, \Sigma^nZ]$, for each pointed CW-complex $X$. My question is why by setting just $E^n(X)=[X,Z^n]$ does not suffice? In his book M.R. Adhikari – Basic Algebraic Topology, seems that defines (on pg.$482$) for a given $\Omega$-spectrum $Z$, the associated (reduced) cohomology theory to be the sequence of functors $\{ [-,Z^n] \}_{n \in \mathbb{Z}}$. Why Switzer does not just take this construction and makes it more complicated?
All I can think, is that Switzer takes any (not necessarily $\Omega$) spectrum and creates a generalized cohomology theory with his construction. Though for any given spectrum, it can be turned into an $\Omega$-spectrum hence I don't get the point.
Best Answer
As you say Switzer defines a cohomology theory $E^\ast$ for any spectrum $E =(E_n,\epsilon_n)$. If $E$ is an $\Omega$-spectrum, then in fact $E^n(X) = [X,E_n]$ which is (in a sense) simpler than the general definition of $E^n(X)$ as the set of homotopy classes of maps of spectra from the suspension spectrum of $X$ to the spectrum $\Sigma^nE$. And yes, you may restrict to $\Omega$-spectra, but why should you reject the more universal approach?
Consider for example the sphere spectrum $S^0$. This is far from being an $\Omega$-spectrum, but the cohomology theory associated to $S^0$ is known as stable cohomotopy. This cohomology theory is well-established in algebraic topology.