Let $C$ be a triangulated category that is closed with respect to arbitrary small coproducts; let $D$ be some class of objects of $C$. Then it would be natural to say that $D$ generates $C$ either if
(i) There are no proper triangulated subcategories of $C$ that are closed with respect to small coproducts and contain $D$.
(ii) For any non-zero object $c$ of $C$ there exist an integer $i$, a $d\in D$, and a non-zero $C$-morphism from $d[i]$ to $c$.
My question is: are there any 'well-known' relations between conditions (i) and (ii)? I only know a proof of their equivalence in the case when $D$ is a set of compact objects.