Categorical Krull-Remak-Schmidt for triangulated categories

Let R be a commutative ring If C₁ and C₂ are R-linear triangulated categories then we can give an obvious triangulated structure on C = C₁ ⊕ C₂ where HomC(U, V) = 0 if U ∈ Cᵢ and V ∈ Cⱼ with i ≠ j. We say a R-linear triangulated category C is disconnected if C = C₁ ⊕ C₂ where Cᵢ are non-zero triangulated subcategories of C. Let Cᵢ and Dⱼ be connected triangulated R categories with i ∈ Γ and j ∈ Λ. Suppose there is an equivalence of triangulated R-categories \[ Φ i ∈ ΓC_i {} j ∈ ΛD_j \] Then we show that there is a bijective function π Γ → Λ such that we have an equivalence Cᵢ Dπ(i) for all i ∈ Γ. We give several examples of connected triangulated categories and also of triangulated subcategories which decompose into utmost finitely many components.