Lemma 13.36.4. Let $\mathcal{D}$ be a triangulated category. Let $E, K$ be objects of $\mathcal{D}$. The following are equivalent
$\mathop{\mathrm{Hom}}\nolimits (E, K[i]) = 0$ for all $i \in \mathbf{Z}$,
$\mathop{\mathrm{Hom}}\nolimits (E', K) = 0$ for all $E' \in \langle E \rangle $.
Proof. The implication (2) $\Rightarrow $ (1) is immediate. Conversely, assume (1). Then $\mathop{\mathrm{Hom}}\nolimits (X, K) = 0$ for all $X$ in $\langle E \rangle _1$. Arguing by induction on $n$ and using Lemma 13.4.2 we see that $\mathop{\mathrm{Hom}}\nolimits (X, K) = 0$ for all $X$ in $\langle E \rangle _ n$. $\square$
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