I'm working out of Sheaves in geometry and logic, for reference.

There is a characterisation of flat functors $A:C \to Set$ as those such that the Grothendieck construction $\int_C A$ is a filtering category. There are more general versions of this result, in which $Set$ is replaced by a more general topos. One should also be able to characterise those discrete opfibrations that arise from flat functors (up to iso/equiv?) - I presume this is written down somewhere? How about if we replace $C$ by an internal category, in a topos $E$ say? Then functors out of $C$ are replaced by discrete opfibrations over $C$ in $E$.

My question is this:

What sort of thing should be considered as the analogue of a flat functor in the internal setting?

I'm working out of Sheaves in geometry and logic, for reference.

There is a characterisation of flat functors $A:C \to Set$ as those such that the Grothendieck construction $\int_C A$ is a filtering category. There are more general versions of this result, in which $Set$ is replaced by a more general topos. One should also be able to characterise those discrete opfibrations that arise from flat functors (up to iso/equiv?). How about if we replace $C$ by an internal category, in a topos $E$ say? Then functors out of $C$ are replaced by discrete opfibrations over $C$ in $E$.

My question is this:

What sort of thing should be considered as the analogue of a flat functor in the internal setting?