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 $$a \xrightarrow{f} b \xrightarrow{g} c$$. 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 $$g$$.