Homological Algebra/Sequences

Lemma: In an $Ab$-enriched category, if $f$ is a kernel, $g$ is a cokernel of $f$, then $f$ is a kernel of $g$.

Corollary: In an abelian category, consider a sequence $a\stackrel{f}{\to }b\stackrel{g}{\to }c$. The following conditions are equivalent:

1. $g$ is a cokernel of $f$ and $f$ is a kernel of $g$.
2. $f$ is a monomorphism and $g$ is a cokernel of $f$.
3. $g$ is an epimorphism and $f$ is a kernel of $g$.