We work in the setting of an abelian category.
We define a monomorphism $S\to B$ allows $A\to B$ if $A\to B$ factors through it.
The title is a lemma in Peter Freyd's abelian categories, P42. The only if part is easy to prove, for proving the if part, say the cokernel of $S\xrightarrow{s}B$ kills $A\xrightarrow{f}B$, then I can find a morphism from $cok(f)$ to $cok(s)$ by using the universal property of cokernel. How do I construct a map from $A$ to $S$?
I would proceed in a different way: using the fact that $$\text{coker}(s) \circ f=0$$ we know that $f$ factors through $\ker (\text{coker} (s))$, but for any monomorphisms $m$, in an abelian category, $\ker(\text{coker}(m))=m$.
The fact on monomorphisms follows from the fact that in an abelian category every monomorphism $m=\ker h$ for some morphism $h$ and that in any additive category $\ker (\text{coker} (\ker f))=\ker f$.