Let $\mathcal{D}$ be a triangulated category and $\mathcal{C}$ be a triangulated subcategory, let $Mor_\mathcal{C}$ be the class of morphisms which have cone in $\mathcal{C}$.
In page 90~91, we have
$\textbf{Lemma 2.1.32}$, it states that a roof $(\alpha,g)$ will be invertible if and only if there exist morphisms $f$ and $h$ in $\mathcal{D}$ so that $g\circ f$ and $h\circ g$ are both in $Mor_\mathcal{C}$.
For the "$\Rightarrow$" part proof, we can deduce that $(id_P,g)$ is invertible from the given condition. Then the author gives the construction of $f$, and says that the statement being dual. However, in constructing $h$, I cannot find any more useful clue except the commutative diagram below, where we can deduce $m\circ l$ is in $Mor_\mathcal{C}$. (Greek alphabet means the denoted morphism is in $Mor_\mathcal{C}$.)
I will appreciate it a lot if anyone can favour to tell me how to find the $h$.
If it helps, you can find that book by searching "triangulated category neeman" in google.
Finally I have an answer from an expert in triangulated categories:
Since $(\text{id}\circ\mu,m\circ l)\sim(\text{id}_P,\text{id}_P)$, we have $m\circ l\in Mor_{\mathcal{C}}$.
We complete the span $(\gamma,m)$ to a square: $$\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.}$$ \begin{array}{ccc} Q&\xrightarrow{m}&P\\ \da{\gamma}&&\da{\beta}\\ Y&\xrightarrow{h}&V \end{array}
where $\beta\in Mor_{\mathcal{C}}$.
Now use two-out-of-three property of set $\{h\circ g\circ\mu, h\circ g,\mu\}$ (where the first element and the third element are in $Mor_{\mathcal{C}}$, we are done.