I don't this this works: if (g .g') is in the domain of the partial functor and g is not, then g will be sent to 0 by the extension of the partial functor, hence g.g' will also be sent to 0 which shouldn't be the case because it is in the domain of the functor. Also it is not entirely clear to me if the original question wanted functor defined on a full subcategory or on an arbitrary sub-category. But in both case, my opinion would be that the only reasonable notion is to talk about "partial natural transformations" which are defined on a subcategory included in the domain of definition of both functor (and hence are natural transformation between ordinary functor). Best wishes, Simon Henry.
The previous suggestion of considering functors to D + 1 was a false start for reasons Fred and Uwe pointed out, but it suggests a better approach: consider functors to the category D~ formed from D by freely adjoining a zero object. Arrows not in S now have somewhere to go (the zero arrow with the appropriate source and target).
I think at the one-categorical level, taking Hom(C,D) to be the zero-preserving functors from C~ to D~, and letting C and D range over all small categories gives a category isomorphic to that of small categories with partial functors as arrows.
Natural transformations between (zero-preserving) functors from C~ to D~ would then give a reasonable notion of partial natural transformations. It certainly captures some, at least, of the natural transformations "more partial" than their source functor, since there will be a zero natural transformation between any two partial functors, corresponding to a "defined nowhere" partial natural transformation when zero-ness is interpreted as undefined as it was in the correspondence between zero-preserving functors from C~ to D~ and partial functors from C to D.
I'm not sure how this fits with the restrictions Robin points out. It seems to allow more partial natural transformations than Robin's observation, since zero arrows can fill in whenever the image object under either the source or target functor is undefined, a partial natural transformation to be a natural transformation between the restrictions of the two partial functors to the intersections of their domain of definition (or a subcategory thereof).
Best Thoughts, David Yetter
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