Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, and suppose that there exists an additive equivalence $F: \mathcal{C} \to \mathcal{D}$. Suppose further that $\mathcal{C} = \text{add}(C\_1 \oplus \dots \oplus C\_n)$ and $\mathcal{D} = \text{add}(D\_1 \oplus \dots \oplus D\_n)$, where the $C\_i$ and $D\_i$ are indecomposable, where here $\text{add}$ denotes additive closure. Finally, suppose that the translation functors on $\mathcal{C}$ and $\mathcal{D}$ permute the $C\_i$ and $D\_i$ respectively, that $F(C\_i) = D\_i$ for all $i$, and that $F(C\_i[1]) = D\_i[1]$ for all $i$. 



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> Does it follow that $F$ is in fact a triangulated equivalence? If not, are there any reasonable hypotheses one can place on $\mathcal{C}$ and $\mathcal{D}$ to ensure that it is?
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Edit: A related question is the following. Suppose that $\mathcal{C} = \text{add}(C\_1 \oplus \dots \oplus C\_n)$ is a triangulated category admitting two triangulated structures, with shifts $[1]$ and $[1]'$, and where the $C\_i$ are indecomposable. Suppose that these shifts permute the $C\_i$ and that $C\_i [1] = C\_i[1]'$ for all $i$. Does it follow that the triangulated structures coincide?



Is this additive equivalence a triangulated equivalence?
No, just take any category equipped with a shift functor satisfying you condition on indecomposables and which admits more than one triangulated structure, and $F$ the identity functor. If you want an explicit example, consider the category of finitely gnerated modules over the dual numbers $k[\epsilon]/(\epsilon^2)$, for $k\neq\mathbb F\_2$. This category is $\operatorname{add}(k,k[\epsilon]/(\epsilon^2))$. The shift functor
is the change of coefficients along the $k$-algebra automorphism defined by $\epsilon\mapsto-\epsilon$. The triangulated structures on this category with shift are in bijection with the set of units $k^\times$.


