Re: Fundamental Theorem of Category Theory?

Apropos of the Yoneda Lemma, is there some reason why it is usually stated on its own rather than as one direction of a characterization of categories of presheaves on J? Unless I've overlooked or misunderstood something it seems to me that the Yoneda Lemma should state that C is a category of presheaves on J if and only if there exists a full, faithful, and dense functor from J to C. This should generalize the characterization of an Archimedean field as any dense extension of the rationals. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]