Consider an additive functor $F \colon C → D$ of abelian categories preserving monos. As Martin has demonstrated here, $F$ may not be left exact.
If $F$ even preserves kernels, $F$ is left exact. But for a functor of abelian categories, preserving kernels is strictly stronger than preserving monos. Are there other additional criterions, not strictly stronger than preservering monos, that would make such an additive, mono-preserving functor left exact?