ct.category theory - Profunctors corresponding to "partial functors"

One thing along these lines that you can say is that if D has, and G preserves, finite limits (or more generally is flat), then so does $hat{G}$ considered as a cocontinuous functor $[D^{op},Set] to [E^{op},Set]$. Since $hat{F}^dagger : [C^{op},Set] to [D^{op},Set]$ is just precomposition with F, it preserves all limits and colimits; thus $hat{G} hat{F}^dagger$ preserves finite limits as soon as G does, without any hypothesis on F. I don't know whether this can be extended to other kinds of limits.

Regarding the more general question of whether $hat{G} hat{F}^dagger$ could be considered a "partial functor," another way to describe it is as the left Kan extension of the composite $D overset{G}{to} E hookrightarrow [E^{op},Set]$ along F. If F is fully faithful, then such an extension is an honest extension, i.e. it restricts back along F to the original functor. So one could think of it as obtained by extending G to objects not in D in the most universal way possible: it maps an object $cin C$ to the formal colimit (viewing $[E^{op},Set]$ as the free cocompletion of E) over all approximations to c by objects of D.

On the other hand, every profunctor can be obtained as a composite $hat{G} hat{F}^dagger$ for some functors G and F (not necessarily an embedding): let the intermediate category D be the two-sided discrete fibration corresponding to that profunctor. An arbitrary profunctor can be thought of as a "generalized functor," but usually not specifically a "partial functor." However, perhaps faithfulness, or full-and-faithfulness, of F implies some properties of the resulting profunctor which makes it seem more like a "partial functor."