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if and only if the rightmost map (from to ) has kernel ; that is, if and only if that map is a monomorphism (injective, or one-to-one).
* Consider the dual sequence . The kernel of the rightmost map is . Therefore the sequence is exact if and only if the image of the leftmost map (from to ) is all of ; that is, if and only if that map is an epimorphism (surjective, or onto).
* Therefore, the sequence is exact if and only if the map from to is both a monomorphism

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