at.algebraic topology - Five lemma in HoTop* and arbitrary pointed model categories

Let $textbf{HoTop}^*$ be the homotopy category of pointed topological spaces. In the following, the word "isomorphism" shall always mean isomorphism in $textbf{HoTop}^*$, i.e. pointed homotopy equivalence. All constructions like cone or suspensions are pointed/reduced.

A triangle $Xto Yto Zto Sigma X$ is called distinguished if it is isomorphic in $textbf{HoTop}^*$ to a triangle of the form $Xstackrel{f}{to} Yhookrightarrowtext{C}ftoSigma X$, where $text{C}ftoSigma X$ is the map collapsing $Y$ to a point.

Problem:

Let $ matrix{X & to & Y & to & Z & to & Sigma Xcrdownarrowalpha &&downarrowbeta&&downarrowgamma &&downarrow&Sigmaalphacr X^{prime} & to & Y^{prime} & to & Z^{prime} & to & Sigma X^{prime}} $ be a morphism of distinguished triangles such that $alpha$ and $beta$ are isomorphisms. Is it true that $gamma$ is an isomorphism, too?

Suggestions:

For a morphism of triangles as above (where $alpha$ and $beta$ are not necessarily isomorphisms), the morphism $gamma^*: [Z^{prime},-]to [Z,-]$ is equivariant with respect to $[Sigmaalpha]^*: [Sigma X^{prime},-]to [Sigma X,-]$. (edit: this is wrong -- see below) Therefore, I thought one could apply theorem 6.5.3 in Hoveys book on Model Categories. Unfortunately, there seems to be a gap at the end of the proof, as already pointed out here.

Therefore, I have the following

Questions:

(1) Am I misunderstanding something in Hovey's proof of 6.5.3(b), or is there really a gap in it? If it is a gap: Do you have any suggestions on how to fix the proof?

(2) If the proof can't be fixed in this generality: Do you have suggestions on how to prove the statement above only for $textbf{HoTop}^*$?

Edit:

(1) The usual proof of this fact for triangulated categories does not work here, because there one uses the fact that $[X,-]$ is abelian-group valued for any $X$ and uses the classical five lemma together with Yoneda to conclude that $gamma$ is an isomorphism. This doesn't seem to work here.

(2) Since partial morphisms of distinguished triangles in $textbf{HoTop}^*$ can always be completed to morphisms of triangles, we can reduce to the case where $alpha$ and $beta$ both equal the identity. Therefore, we have a commutative diagram (in $textbf{HoTop}^*$, i.e. a homotopy commutative diagram in $textbf{Top}^*$)

$matrix{X & to & Y & to & Z & to & Sigma Xcrdownarrow & text{id}_X &downarrow & text{id}_Y&downarrow&gamma&downarrow&text{id}_{Sigma X}cr X & to & Y & to & Z& to & Sigma X}$

and we have to prove that $gamma$ is a homotopy equivalence.

Hovey's proof

The way Hovey proceeds in his proof is as follows: We know the following things:

(1) $gamma^*: [Z,-]to [Z,-]$ is $[Sigma X,-]$-equivariant

(2) Two maps $c,din[Z,W]$ are equal in $[Y,W]$ if and only if they lie in the same $[Sigma X,W]$-orbit.

From (2) and the commutativity of the middle square it follows that for any $hin [Z,W]$ there is some $rhoin[Sigma X,W]$ such that $gamma^*(h)=h.rho$; in other words $gamma^*$ doesn't change the $[Sigma X,-]$-orbit.

Now, suppose there are $g,hin [Z,W]$ such that $gamma^*(h)=gamma^*(g)$. Then, again by the commutativity of the middle square, there is some $alphain [Sigma X,W]$ such that $g = h.alpha$. Thus, by (1), $gamma^*(g) = gamma^*(h).alpha = gamma^*(g).alpha$, and so $alphaintext{Stab}(gamma^*(g))$.

The point is that Hovey now wants to show that $text{Stab}(gamma^*(g))=text{Stab}(g)$; this would imply $alphaintext{Stab}(g)$, and thus $h = g.alpha^{-1} = g$ as required. The inclusion $text{Stab}(gamma^*(g))supsettext{Stab}(g)$ is obvious. For the other inclusion, I have no idea how to prove it.

Do you see how one can fix the proof?

FINAL EDIT

I made a mistake in proving that for any morphism of triangles $(alpha,beta,gamma)$ the morphism $gamma^*$ is equivariant with respect to $(Sigmaalpha)^*$. This is wrong.

So what remains is the question on how to fix the proof of theorem 6.5.3 in Hovey's book. Any suggestions?

Thank you.