Let $\mathbf{C}$ be a pointed category, that is to say, a category with a zero object $0$, and suppose that $\mathbf{C}$ has all kernels and cokernels, and suppose also that every monomorphism in $\mathbf{C}$ is a kernel. $\DeclareMathOperator{coker}{coker}$
Let us say that a morphism $g$ is a pseudo-epimorphism if $l \circ g = 0$ implies $l = 0$: so $g$ is a pseudo-epimorphism if and only if $\coker g = 0$. Consider a morphism $f : A \to B$; let $k = \ker (\coker f)$, the regular image of $f$. Since $\coker(f) \circ f = 0$, $f$ must factor through $k$, say $f = k \circ g$.
Question. Under these hypotheses, why is $g$ pseudo-epic?
Examples of categories satisfying these hypotheses: the category of pointed sets $\textbf{Set}_*$, the opposite category of groups $\textbf{Grp}^\textrm{op}$, and of course any abelian category. The proof in the case of abelian categories is reasonably straightforward, due to the presence of sufficient colimits and exactness conditions: indeed, if $l \circ g = 0$, we take the pushout $\tilde{k}$ of $l$ along $k$, since $\coker k = \coker f$, we find that $\tilde{k} \circ l = 0$, and $\tilde{k}$ is monic since $k$ is, so $l = 0$ as required. Unfortunately, $\mathbf{C}$ does not have pushouts in general...