The Stacks project

Remark 13.35.5. Let $F : \mathcal{T} \to \mathcal{T}'$ be an exact functor of triangulated categories. Given a full subcategory $\mathcal{A}$ of $\mathcal{T}$ we denote $F(\mathcal{A})$ the full subcategory of $\mathcal{T}'$ whose objects consists of all objects $F(A)$ with $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. We have

\[ F(\mathcal{A}[a, b]) = F(\mathcal{A})[a, b] \]
\[ F(smd(\mathcal{A})) \subset smd(F(\mathcal{A})), \]
\[ F(add(\mathcal{A})) \subset add(F(\mathcal{A})), \]
\[ F(\mathcal{A} \star \mathcal{B}) \subset F(\mathcal{A}) \star F(\mathcal{B}), \]
\[ F(\mathcal{A}^{\star n}) \subset F(\mathcal{A})^{\star n}. \]

We omit the trivial verifications.

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