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The following, if true, should probably be "standard," but I don't know where to look. I'd rather be slightly imprecise about hypotheses in the hope that there's a good general answer. Feel free to tell me that somewhat stronger hypotheses are needed.

Let $D$ be a pretriangulated dg category equipped with a $t$-structure for which the heart is an abelian category $A$. Let $T\subset A$ be a localizing subcategory, so in particular the quotient abelian category $A/T$ exists. Let $C\subset D$ be the full subcategory of $D$ consisting of objects whose cohomologies (with respect to the given $t$-structure) lie in $T$.

Belief: $A/T$ is equivalent to the heart of $D/C$ (w.r.t. induced $t$-structure).

Under what hypotheses is it true? What's a reference?

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Note that I would be happy, for a start, with a reference for the case when $D = D(A)$ is (a dg enhancement of) the derived category of $A$. –  Thomas Nevins May 11 '11 at 14:32

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