Categorical Krull-Remak-Schmidt for triangulated categories

Let $R$ be a commutative ring If ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C}={\mathcal{C}}_1\oplus {\mathcal{C}}_2$ where $Ho{m}_{\mathcal{C}}(U,V)=0$ if $U\in {\mathcal{C}}_i$ and $V\in {\mathcal{C}}_j$ with $i\mathrm{\ne }j$. We say a $R$-linear triangulated category $\mathcal{C}$ is disconnected if $\mathcal{C}={\mathcal{C}}_1\oplus {\mathcal{C}}_2$ where ${\mathcal{C}}_i$ are non-zero triangulated subcategories of $\mathcal{C}$. Let ${\mathcal{C}}_i$ and ${\mathcal{D}}_j$ be connected triangulated $R$ categories with $i\in \mathrm{\Gamma }$ and $j\in \mathrm{\Lambda }$. Suppose there is an equivalence of triangulated $R$-categories $$\mathrm{\Phi }\text{\hspace{0.17em}⁣}:\underset{i\in \mathrm{\Gamma }}{⨁}{\mathcal{C}}_i\stackrel{\cong }{\to }\underset{j\in \mathrm{\Lambda }}{⨁}{\mathcal{D}}_j$$ Then we show that there is a bijective function $\pi \text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to \mathrm{\Lambda }$ such that we have an equivalence ${\mathcal{C}}_i\cong {\mathcal{D}}_{\pi (i)}$ for all $i\in \mathrm{\Gamma }$. We give several examples of connected triangulated categories and also of triangulated subcategories which decompose into utmost finitely many components.