# “Third isomorphism theorem” for abelian categories

If we have an exact sequence of modules $O\rightarrow A\stackrel{f}{\rightarrow}B\stackrel{g}{\rightarrow}C$, then from the third isomorphism theorem we can conclude that $\operatorname{Im} g\cong \operatorname{Coker} f$.

I think that this is also true in every abelian category.

We have that

\begin{align} \operatorname{Im}g&=\operatorname{Ker}(\operatorname{Coker}g)\cong \operatorname{Coker}(\operatorname{Ker}g)\cong \operatorname{Coker}( \operatorname{Im} f)\\ &= \operatorname{Coker}(\operatorname{Coker}(\operatorname{Ker} f))\cong \operatorname{Coker}(f). \end{align}

First isomorphism holds by the axiom(s) of abelian categories, second holds by exactness of the sequence $O\rightarrow A\stackrel{f}{\rightarrow}B\stackrel{g}{\rightarrow}C$ at $B$ and the last holds because $f$ is a monomorphism.

Is my reasoning correct?