Adams' proof of Homotopy Extension Theorem for CW-spectra

Question

I'm studying stable homotopy category, but I find the language of model category hard to understand. For example, I was suggested reading this (which I call "MMSS")

Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf).

So I return to the classics and read part III of this book

J. F. Adams, Stable homotopy and generalised homology.

In this, Adams constructed the category of CW-spectra. It is fine until he proved Lemma 3.1, the Homotopy Extension Theorem for CW-spectra.

Let $X,A$ be a pair of CW-spectra, and $Y,B$ be a pair of spectra such that $\pi_{*}(Y,B)=0$. Suppose given a map $f:X\to Y$ and a homotopy $h:\operatorname{Cyl}(A)\to Y$ from $f|_A$ to a map $g: A\to B$. Then the homotopy can be extended over $\operatorname{Cyl}(X)$ so as to deform $f$ to a map $X\to B$.

Here $\operatorname{Cyl}(X)=I^{+}\wedge X$. He claimed that $f$ is represented by a function $f': X'\to Y$ (this is just definition), and $h$ by a function $h':\operatorname{Cyl}(A')\to Y$, where $X'\supset A'$, $X'$ is cofinal in $X$ and $A'$ is cofinal in $A$. Here the existence of $h'$ is questionable, since it is not known whether every cofinal subspectra in $\operatorname{Cyl}(A)$ contains $\operatorname{Cyl}(A')$ for some cofinal subspectra $A'$ of $A$ (I couldn't see it myself). Moreover, the proof relies on an argument using Zorn's Lemma, in which the existence of such $h'$ is indispensable.

A quick search tell me that CW-spectra is now called CW-prespectra, and is a special version of sequential spectra. Also, on nLab (see this) they advised to ignore section $III.3$ in Adams' book and follow the treatment in MMSS, so maybe this part of the book is already outdated. My questions are

1. Is the original proof correct? For example, one could show the existence of $h'$ somehow.
2. If the original proof is wrong, is the theorem still true? Is there a version of it in the modern theory? How can I prove it?

Any help will be appreciated. Also, some advice or sources about this topic would be very kind.

Answer 1

In the comments, I asserted the following.

Proposition. Fix a finite complex $K$ and a spectrum $A$, and let $C$ be a cofinal subspectrum of $A \wedge K_+$. Then $C$ contains a subspectrum of the form $A' \wedge K_+$, where $A'$ is a cofinal subspectrum of $A$.

Proof. For each cell in $K$, let $C_e$ be the intersection of $C$ and $A \wedge e_+$ in $A \wedge K_+$. We may regard $C_e$ as a subspectrum of $A \simeq A \wedge e_+$, and it is cofinal in $A$ since $C$ is cofinal in $A \wedge K_+$. Let $A'$ be intersection of all the $C_e$ as $e$ varies over the cells of $K$. Since $K$ only has finitely many cells, $A'$ is cofinal in $A$, and $A' \wedge K_+$ is a subspectrum of $C$.