I have been reading from the Stacks project, and Lemma 13.5.4. says:
Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of pre-triangulated categories. Let $$ S = \{f \in \text{Arrows}(\mathcal{D}) \mid F(f)\text{ is an isomorphism}\} $$ Then $S$ is a saturated multiplicative system compatible with the triangulated structure on $\mathcal{D}$.
I can't show that MS3 holds in S without using TR4 (specifically, existence of morphism $i$ in following diagram), but TR4 only holds in triangulated categories.
Could anyone explain to me why the diagram commutes without using TR4? Thanks.
\begin{array} 0Z &\stackrel{t}{\longrightarrow}& X &\stackrel{g}{\longrightarrow}& Q &\longrightarrow & Z[1] \\ &&\downarrow{id_x} &&\downarrow{i} \\ &&X &\stackrel{a}{\longrightarrow}& Y \\ &&&& \downarrow{j} \\ &&&& W \end{array}