Lemma 13.36.4. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{D}' \subset \mathcal{D}$ be a full triangulated subcategory. The following are equivalent

1. the inclusion functor $\mathcal{D}' \to \mathcal{D}$ has a left adjoint, and

2. for every $X$ in $\mathcal{D}$ there exists a distinguished triangle

$K \to X \to X' \to K[1]$

in $\mathcal{D}$ with $X' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$ and $\mathop{\mathrm{Hom}}\nolimits (K, Y') = 0$ for all $Y' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$.

Proof. Omitted. Dual to Lemma 13.36.3. $\square$

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