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This is a question for someone who read (or wrote) enough of Lurie's HTT to know a reference. Suppose $D,E$ are small "diagram" $(\infty,1)$-categories, and $\mathcal{C}$ is a stable infinity-category fibered over $D\times E$ (i.e., we have a functor $D\times E$ to the category of stable infinity-categories with stable maps). What are some minimal conditions for the existence of an equivalence $$\text{lim}_D\text{colim}_E\mathcal{C}\cong \text{colim}_E\text{lim}_D\mathcal{C}?$$ In the (simplest) case I'm interested in, $E$ is an ordinary coproduct category, $\bullet \leftarrow \bullet \rightarrow\bullet$ and the part of $\mathcal{C}$ that lives over the middle bullet is the path category of a topological space (obtained by inverting all morphisms in the other index category $D$).

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Under appropriate assumptions one has: Proposition filtered colimits commute with finite limits. And Lemma sifted colimits commute with finite products. – Adeel Khan Aug 13 '14 at 4:11
@Adeel Thanks, but neither of these is quite what I need. I need index diagrams that are more complicated than just products (so isn't enough), and is only about limits/colimits in the category of spaces (I need the category of stable infty-categories). However, there is something about my case that's not that far from the category of spaces, and I'm editing the question accordingly. – Dmitry Vaintrob Aug 13 '14 at 7:22

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