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 $$\text{lim}_D\text{colim}_E\mathcal{C}\cong \text{colim}_E\text{lim}_D\mathcal{C}?$$

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 $D$).