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Suppose we have a span of categories $C \overset{F}{\hookleftarrow} D \overset{G}{\rightarrow} E$, where $F$ is a subcategory embedding. We can lift these normal functors to profunctors $\hat F$ and $\hat G$ and compose the "formal" adjoint $\hat F^\dagger$ with $\hat G$ to obtain a profunctor $\hat G \hat F^\dagger : C \nrightarrow E$. To what extent could one think of this as a partial functor, and what nice behaviours could it inherit from $F$ and $G$? For instance, if $F$ reflects products and $G$ preserves them, does $\hat G \hat F^\dagger$ preserve them where ever it is defined?

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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 $c\in C$ 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."

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I guess this analogy goes through by replacing the notion of "subset where the image of a function is defined" with "subcategory where the image of a profunctor is representable." Interesting that a yes or no question is replaced by a question of approximations. Are there interesting ways one can gauge how good such approximations are? The examples I'm thinking of are categories of matrices over various rigs, especially Mat(C) <--< Mat(R+) -->> Rel. –  Aleks Kissinger Aug 19 '10 at 18:15
    
Well, there are various sorts of "almost-representability" that one can consider, often phrased in terms of a presheaf being a certain sort of limit or colimit of representables. Replacing of yes/no questions by more refined information is quite common in categorification. –  Mike Shulman Aug 20 '10 at 3:13
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