Lemma 13.36.5. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. If $E$ is a classical generator of $\mathcal{D}$, then $E$ is a generator.

**Proof.**
Assume $E$ is a classical generator. Let $K$ be an object of $\mathcal{D}$ such that $\mathop{\mathrm{Hom}}\nolimits (E, K[i]) = 0$ for all $i \in \mathbf{Z}$. By Lemma 13.36.4 $\mathop{\mathrm{Hom}}\nolimits (E', K) = 0$ for all $E'$ in $\langle E \rangle $. However, since $\mathcal{D} = \langle E \rangle $ we conclude that $\text{id}_ K = 0$, i.e., $K = 0$.
$\square$

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