I follow the treatment on page 178 in the book categories and sheaves by Kashiwara/Schapira (which is inspired by the great answer here).
Let $\mathcal{A}$ be an abelian category. Consider a complex $$X \overset{f}{\to} Y \overset{g}{\to} Z \qquad g \circ f = 0$$ in $\mathcal{A}$. Define $$\mathrm{im}f := \mathrm{ker}(\mathrm{coker}f).$$ Now we have the following statement:
Since $\mathrm{im}f \to Y \to Z$ is zero, $\mathrm{im}f \to Y$ factors through $\mathrm{ker}g$, i.e. we have $\varphi : \mathrm{im}f \to \mathrm{ker}g$. This $\varphi$ is a monomorphism.
I do not quite see, how we can deduce the existence of $\varphi$ (and moreover it should be unique I think). I tried to approach this by considering $$g \circ \mathrm{im} f$$ Since if this is $0$, we would get that $\mathrm{im}f$ is a fork for $f$ and $0$, hence since $\mathrm{ker}g$ is a kernel, we have that there is a unique map $$\mathrm{im}f \to \mathrm{ker}g.$$ I think this is plausible, since $g \circ f = 0$ holds. However: I am unable to show this in the setting of abelian categories and that this morphism is also monic. Any help is appreciated.
Edit. It turned out, that the existence and uniqueness is easy to prove, however, I have not managed yet to show the monic property.
