kernel and monomorphism in pointed category

Let \(\mathcal{A}\) be a pointed and \(f: X \rightarrow Y\) a morphism

TFAE:

1. \(f\) is a monomorphism
2. the kernel is the zero object: \(\mathrm{ker}(f) = 0\)

follows dually to cokernel and epimorphism in an abelian category