Suppose I have an additive category $\mathcal{C}$ and a pair of composable arrows:
$$A \longrightarrow B \longrightarrow C.$$
It makes no sense to ask if this sequence is exact at $B$ since the category $\mathcal{C}$ doesn't have kernels or images.  However: the Yoneda embedding produces a sequence of functors
$$\mbox{Hom}(A,-) \longleftarrow \mbox{Hom}(B,-) \longleftarrow \mbox{Hom}(C,-)$$
and it *does* make sense to ask if this sequence is exact since the functor category $\[\mathcal{C},\mbox{Ab}\]$ is abelian.

Similarly we may ask if
$$\mbox{Hom}(-,A) \longrightarrow \mbox{Hom}(-,B) \longrightarrow \mbox{Hom}(-,C)$$
is an exact sequence.  So here are my questions:

 1. What are criteria for determining if a given composable pair of arrows become exact under one of the two Yoneda embeddings?
 2. What properties of $\mathcal{C}$ guarantee that the co- and contravariant Yoneda embeddings agree on which sequences become exact?