Definition 13.5.1. Let $\mathcal{D}$ be a pre-triangulated category. We say a multiplicative system $S$ is *compatible with the triangulated structure* if the following two conditions hold:

For $s \in S$ we have $s[n] \in S$ for all $n \in \mathbf{Z}$.

Given a solid commutative square

\[ \xymatrix{ X \ar[r] \ar[d]^ s & Y \ar[r] \ar[d]^{s'} & Z \ar[r] \ar@{-->}[d] & X[1] \ar[d]^{s[1]} \\ X' \ar[r] & Y' \ar[r] & Z' \ar[r] & X'[1] } \]whose rows are distinguished triangles with $s, s' \in S$ there exists a morphism $s'' : Z \to Z'$ in $S$ such that $(s, s', s'')$ is a morphism of triangles.

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