Let $\mathcal{A}$ be a category. If $\mathcal{A}$ is pointed, i.e. has zero objects, then $f$ monic $\implies \operatorname{ker} f = 0$. If $\mathcal{A}$ is abelian, we have the equivalence $f$ monic $\iff \operatorname{ker} f = 0$. However this is true in $\mathsf{Gp}$, which is not abelian. When in general (or more generally) is this true?
It is true when $\mathcal{A}$ is conormal, or more generally when the coequaliser of any two morphisms is a cokernel, but do we need this?