It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are all sorts of good reasons for this (Yoneda's lemma being a big one, and the fact that this is a topos). There are other versions, sometimes you look at contravariant functors into some other category, like groups (e.g. for algebraic groups). But I have never seen the dual notion: covariant functors from some other category INTO $C$.

Is this notion as useful as the notion of a presheaf? I guess it's like looking at a category "over" $C$ as opposed to a category "under" it. I guess, hidden in this question is the question: Can one learn anything about a category $C$ by looking at presheaves of $C$? For example, does the difference between presheaves of sets and presheaves of ableian groups tell us anything about the differences between the category of Groups and the category of Sets?

It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are all sorts of good reasons for this (Yoneda's lemma being a big one, and the fact that this is a topos). There are other versions, sometimes you look at contravariant functors into some other category, like groups (e.g. for algebraic groups). But I have never seen the dual notion: covariant functors from some other category INTO $C$.

Is this notion as useful as the notion of a presheaf? I guess it's like looking at a category "over" $C$ as opposed to a category "under" it. I guess, hidden in this question is the question: Can one learn anything about a category $C$ by looking at presheaves of $C$? For example, does the difference between presheaves of sets and presheaves of ableian groups tell us anything about the differences between the category of Groups and the category of Sets?

EDIT: There seems to be a bit of confusion of what I mean by "dual of a presheaf." I don't mean "copresheaves" (which I didn't know existed before I asked this), I mean what you get by reversing the arrow of the functor not taking the opposite category. So I'm looking at functors into the category of interest, as opposed to out of them. I can see how this is confusing because usually "dual" doesn't turn around functors, just arrows inside categories. So I guess I mean presheaves in $C^{op}$... (but covariant instead of contravariant?)... who knows.