I encountered the following HW level problem:
Assume $\mathcal{C}$ is a category which admits a zero object and kernels (by this word I think the author means equalizers). Prove that a morphism $\mathbb{}f:X \rightarrow Y$ is a monomorphism iff $\mathbb{} Kerf \simeq 0$.
So $\Rightarrow$ is obvious, while I have no idea why $\Leftarrow$ should work. Any ideas? General comments are also welcome!
P.S. According to Mitchell's "Theory of categories" pg15, $\Leftarrow$ is not true in general ... so seeing counterexamples would be delightful. Seems that we should start from a category which is not normal....
P.S.2 These are actually from Schapira's notes:link Pg44. Alas, Mitchell vs Schapira!