Definition 13.3.3. Let $\mathcal{D}$, $\mathcal{D}'$ be pre-triangulated categories. An exact functor, or a triangulated functor from $\mathcal{D}$ to $\mathcal{D}'$ is a functor $F : \mathcal{D} \to \mathcal{D}'$ together with given functorial isomorphisms $\xi _ X : F(X[1]) \to F(X)[1]$ such that for every distinguished triangle $(X, Y, Z, f, g, h)$ of $\mathcal{D}$ the triangle $(F(X), F(Y), F(Z), F(f), F(g), \xi _ X \circ F(h))$ is a distinguished triangle of $\mathcal{D}'$.

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