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 $X\to Y\to Z\to \Sigma X$ is called *distinguished* if it is isomorphic in $\textbf{HoTop}^*$ to a triangle of the form $X\stackrel{f}{\to} Y\hookrightarrow\text{C}f\to\Sigma X$, where $\text{C}f\to\Sigma X$ is the map collapsing $Y$ to a point.

**Problem:**

Let $\ \ \matrix{X & \to & Y & \to & Z & \to & \Sigma X\cr\downarrow\alpha &&\downarrow\beta&&\downarrow\gamma &&\downarrow&\Sigma\alpha\cr 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 $[\Sigma\alpha]^*: [\Sigma X^{\prime},-]\to [\Sigma X,-]$. 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][1].

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 $\textbf{HoTop}^*$?

You can find the details on triangles in $\textbf{HoTop}^*$ I already worked out on my [webpage][2] (the above question is part of a big "christmas-exercise" for the topology III lecture that should introduce the students to triangulated categories).

Thank you.

  [1]: http://mathoverflow.net/questions/11977/equivariant-map-preserves-stabilizer
  [2]: http://www.uni-bonn.de/~habecker/dokumente/tutorium/ExTopIII/Sheet10.pdf