David, My guess would be to start with the tricategory of spans of categories: The objects are categories, and a 1-morphism from C to D is category E with two functors, F:E \to C and G:E \to D, and composition is given by computing 2-categorical pullbacks of categories (so is associative only up to equivalence), 2-morphisms between a span (F,G):E \to C \times D and a span (F',G'):E' \to C \times D are given by a functor H:E \to E' together with natural isomorphisms making everything commute, and 3-morphisms between two such 2-morphisms H,H':E \to E' (I'm suppressing the natural isomorphisms in my notation, but they're still there) are natural transformations compatible with the natural isomorphisms associated to H and H'. Now a partial functor from C to D is a particular case of a span, F:E \to C and G:E \to D, but where F is required to be full and faithful. Since Span(Cat) is a tricategory, Hom_Span(Cat)(C,D) is a bicategory. Take the full sub-bicategory of Hom_Span(Cat)(C,D) on those spans (F,G) where F is full and faithful. This is a bicategory of partial functors. -Dave ________________________________________ From: David Leduc [david.leduc6@googlemail.com] Sent: Monday, November 07, 2011 1:55 PM To: categories Subject: categories: Partial functor Hi, A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]