Remark 13.35.6. Let $\mathcal{T}$ be a triangulated category. Given full subcategories $\mathcal{A}_1 \subset \mathcal{A}_2 \subset \mathcal{A}_3 \subset \ldots $ and $\mathcal{B}$ of $\mathcal{T}$ we have

\[ \left(\bigcup \mathcal{A}_ i\right)[a, b] = \bigcup \mathcal{A}_ i[a, b] \]

\[ smd\left(\bigcup \mathcal{A}_ i\right) = \bigcup smd(\mathcal{A}_ i), \]

\[ add\left(\bigcup \mathcal{A}_ i\right) = \bigcup add(\mathcal{A}_ i), \]

\[ \left(\bigcup \mathcal{A}_ i\right) \star \mathcal{B} = \bigcup \mathcal{A}_ i \star \mathcal{B}, \]

\[ \mathcal{B} \star \left(\bigcup \mathcal{A}_ i\right) = \bigcup \mathcal{B} \star \mathcal{A}_ i, \]

\[ \left(\bigcup \mathcal{A}_ i\right)^{\star n} = \bigcup \mathcal{A}_ i^{\star n}. \]

We omit the trivial verifications.

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