*Notation:* Let $\mathcal T$ be a triangulated category, and let $\mathcal E$ be a full subcategory of $\mathcal T$. I write $\langle \mathcal E \rangle$ to indicate the smallest strictly full triangulated subcategory of $\mathcal T$ which contains $\mathcal E$.

Now, assume we are given triangulated categories $\mathcal T= \langle \mathcal E \rangle$ and $\mathcal T' = \langle \mathcal E' \rangle$, such that the "generating subcategories" $\mathcal E$ and $\mathcal E'$ are equivalent (or even isomorphic). Is it true that $\mathcal T$ is equivalent to $\mathcal T'$? Or, if not in general, is it true under some reasonable assumptions?