Let $E$ and $F$ be CW-spectra and $\tilde{F}$ be cofinal. Let $f \colon E \to F$ be a function of spectra. I am trying to prove: There is $\tilde{E} \subseteq E$ cofinal such that $f$ maps $\tilde{E}$ into $\tilde{F}$. See in Adams Blue Book III Lemma 2.6.(i).
My idea so far is to define a subspectrum $\tilde{E}$ such that $\tilde{E}_n$ is the disjoint union of the cells $e$ of $E_n$ with $f_n(e) \subseteq \tilde{F}$. At first, $\tilde{E}_n$ has to be a subspace of $E_n$, but this should be clear since $E_n$ is the disjoint union of its cells. But how do you show that this is a spectra, that means that the map $\Sigma E_n \to E_{n+1}$ maps $\Sigma \tilde{E}_n $ to $\tilde{E}_{n+1}$, and is my idea right so far?
To show that $\sigma_n: \Sigma E_n \to E_{n+1}$ restricts to $\Sigma \tilde{E}_n \to \tilde{E}_{n+1}$ we must show that $f_{n+1}$ maps $\sigma_n(\Sigma\tilde{E}_n)$ into $\tilde{F}_{n+1}$ (because then by definition $\sigma_n(\Sigma\tilde{E}_n) \subset \tilde{E}_{n+1}$). That is, we must show that $f_{n+1}(\sigma_n(\Sigma\tilde{E}_n)) \subset \tilde{F}_{n+1}$.
Since $f$ is a function of spectra, $f_{n+1}(\sigma_n(\Sigma \tilde{E}_n)) = \tau_n(\Sigma f_n(\Sigma\tilde{E}_n))$ (here $\tau_n$ is a structure map of $F$). The latter is indeed a subset of $\tilde{F}_{n+1}$, since $f_n$ maps $\tilde{E}_n$ into $\tilde{F}_n$, so $\Sigma f_n$ maps $\Sigma \tilde{E}_n$ into $\Sigma \tilde{F}_n$, and so since $\tilde{F}$ is a subspectrum $\tau_n$ maps $\Sigma \tilde{F}_n$ into $\tilde{F}_{n+1}$.