In Adams' definition, a map $f: X \to Y$ of CW spectra consists of a cofinal subcomplex $X'\subseteq X$ and maps $f_n: X'_n \to Y_n$ that commute with the structure maps. This definition is reproduced in Hatcher's notes. But modern categories of spectra (such as these) never take the step of passing to a cofinal subcomplex. So how do the moderns manage to end up with the same homotopy category as Adams?
Adams motivates this step by pointing out that there should be a Hopf map $\Sigma \mathbb{S} \to \mathbb{S}$, but that this map can't be defined in degrees 0 and 1. So a sharper formulation of the question would be: how do modern categories of spectra manage to represent the Hopf map?
Ok, I think I see this now. The point is that the inclusion of a cofinal subcomplex is a stable homotopy equivalence. This strikes me as weird: by Whitehead's theorem for spectra, the cofinal subcomplex inclusion must have a homotopy inverse. What on earth does it look like?
Anyway, here's an argument. A cofinal subcomplex is $X'\subseteq X$ such that for every cell $e^d \subseteq X_n$, there exists $k$ such that $\Sigma^k e^d$ is mapped into $X'_{n+k}$ by the canonical composite of (suspensions of) structure maps $\Sigma^k X_n \to X_{n+k}$. We want to show that for every $i$, $X'\subseteq X$ induces an isomorphism $\pi_i(X') = \varinjlim_n \pi_{i+n}(X_n) \to \pi_i(X) = \varinjlim_n \pi_{i+n}(X)$.
To show this map is surjective, pick an element of $[\phi] \in \pi_i(X)$. It is represented by some map $\phi: S^{i+n} \to X_n$ which we can take to be cellular. By compactness, there are finitely many cells in the image of $\phi$, so by cofinality there exists a $k$ such that $k$-fold suspension of each of these cells is in $X'_{n+k}$. Hence the image of the $\Sigma^k \phi$ is mapped into $X'_{n+k}$, and $\Sigma^k\phi \in \pi_{i+k}(X_{n+k})$ represents a preimage of $[\phi]$ in $\pi_i(X)$.
The argument for injectivity is similar, where the role of $\phi$ is replaced by a nullhomotopy of a map from a sphere, i.e. a map $D^{i+1+n} \to X_n$.