Given a functor $F:A\to B$ of abelian categories we may say that $F$ is left exact if it maps exact sequences to left exact sequences, and similarily for right. For arbitrary categories, we may say that $F$ is left exact if it preserves finite limits (supposedly, this was introduced in SGAIV, but I don't have it). The question is thus: are these definitions equivalent in an abelian category? That this latter definition implies the first is clear to me, but the other gives me more trouble.
Thanks,
Eivind