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I guess I've heard it stated in words -- there is a full embedding of C in Funct(C

^{op}, Set). I know what all of the words mean, but I'm not sure if that helps yet.

There's one particular application of it that interests me, and I can't really see how to use it: there's a proof that shows the existence of a natural transformation Hom(_, A) --> Hom(_, B).

From this, they then conclude (by the Yoneda lemma!) that this natural transformation must come from a uniquely determined map A --> B. (The existence of this map is what interests me)

How does that follow?