The Stacks project

Lemma 13.4.7. Let $\mathcal{D}$ be a pre-triangulated category. Let $f : X \to Y$ be a morphism of $\mathcal{D}$. There exists a distinguished triangle $(X, Y, Z, f, g, h)$ which is unique up to (nonunique) isomorphism of triangles. More precisely, given a second such distinguished triangle $(X, Y, Z', f, g', h')$ there exists an isomorphism

\[ (1, 1, c) : (X, Y, Z, f, g, h) \longrightarrow (X, Y, Z', f, g', h') \]

Proof. Existence by TR1. Uniqueness up to isomorphism by TR3 and Lemma 13.4.3. $\square$


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