Subcategories of the Verdier quotient?

Let $\mathcal T$ be a triangulated category and $\mathcal C$ a thick triangulated subcategory. We consider the Verdier quotient $\mathcal T/\mathcal C$.

Is there a bijective correspondence between thick triangulated subcategories of the quotient $\mathcal T/\mathcal C$ and thick triangulated subcategories of $\mathcal T$ containing $\mathcal C$?

It "feels" true, but I have learned never to take chances with technicalities of triangulated categories.