TITLE: A remark on triangulated categories and localizations in Kashiwara & Schapira's *Sheaves on Manifolds*
QUESTION [8 upvotes]: I'm having a little difficulty understanding the following remark in Kashiwara & Schapira's Sheaves on Manifolds:

Since the term "null system" doesn't appear to be very common, here is the definition, along with its main application:

The notation $\mathcal{C}/\mathcal{N}$ in the remark stands for the localization of $\mathcal{C}$ at $S(\mathcal{N}).$ The functor $Q$ is the canonical functor sending a morphism $f:X \rightarrow Y$ to the roof $X \xleftarrow{1} X \xrightarrow{f} Y.$
Now, I'm fine with one direction. Namely, it's clear enough that if there exists $Y$ such that $X \oplus Y \in \mathcal{N},$ then $Q(X) \cong 0,$ for the triangulated category and null system axioms show that $X \oplus Y$ being in $\mathcal{N}$ is equivalent to the zero morphism $X \oplus Y \rightarrow 0$ being in $S(\mathcal{N}).$ Hence $0 \leftarrow X \oplus Y \rightarrow X$ is a roof in $\mathcal{C}/\mathcal{N},$ so that the diagram

shows that $X$ is isomorphic to $0$ in the localization (here $i,$ resp. $p,$ is the canonical inclusion, resp. projection).
The other direction is still unclear to me. We have a diagram of roofs
,
where $Z \in \mathcal{N}$ and $f \circ g = 1.$ I have tried to show that $Z \cong X \oplus X[1],$ but I don't even think it is true.
Your help would be greatly appreciated!

REPLY [5 votes]: The morphism $g\colon X \to Z$ sits in a distinguished triangle$\require{AMScd}$ $\DeclareMathOperator{Hom}{Hom}$
$$
\begin{CD}
X @>g>> Z @>g'>> Y @>\delta>> X[1]
\end{CD}
$$
The relation $fg = 1_X$ implies that $g$ is monic. Since the composition of consecutive maps in a distinguished triangle is zero, $g \circ (\delta[-1]) = 0$, and this implies that $\delta[-1] = 0$. Therefore $\delta = 0$. 
Now observe that we have a morphism of distinguished triangles
\begin{CD}
X  @>g>> Z @>g'>> Y @>0>> X[1]\cr
@VV{1_{X}}V @VV{[f \; g']^t}V @VV{1_{Y}}V @VV1_{X[1]}V \cr
X  @>[1_X \; 0]^t>> X \oplus Y @>[0\;1_Y]>> Y @>0>> X[1]
\end{CD}
and we deduce from the five lemma for triangulated categories that $[f \; g']^t\colon Z \to X \oplus Y$ is an isomorphism.