categories: Generalized Subfunctors

["Robert W. McGrail" <mcgrail@bard.edu>]

Dear Categorists,

I wish to label the following notion of (for lack of a better term)
"generalized subfunctor" of a functor H: J ---> C in a manner that is
consistent with the categorical community.  What I call a "generalized
subfunctor" of F is a category D, a pair of functors F: C --->D and G: J
---> D, and a monic natural transformation m: G ---> FoH.  My motivation
is to formulate the notion of "generically creating a generalized
subfunctor" to such a functor H.

Is there some standard alternative to the overloaded term
"generalized"?  By the way, I am somewhat interested in the case where H
sends certain J-spans to product diagrams in C, so sketch-theoretic
ideas are certainly welcome.

--
Best Regards,
  Bob McGrail

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By virtue knowledge

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