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?

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 covariant and contravariant Yoneda embeddings agree on which sequences become exact?