Two questions on kernels Let $\mathcal{C}$ be a category with a zero object $0$. Let $B\xrightarrow{\ \ g\ \ } C$ be an arrow with kernel $K\xrightarrow{\ \ k\ \ }B$. Two questions arose, which I were not able to answer:
(1): If $k=0$, does this imply that $g$ is a monomorphism? If not, can you provide a counterexample?
(2): Let $K'\xrightarrow{\ \ k'\ \ }B$ be another arrow. If both $k$ factors through $k'$ and $k'$ factors through $k$, does this imply that $k'$ is a kernel aswell (and therefor the two factorization arrows are mutually inverse)?

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*No. There are more pointed categories out there than abelian categories. For instance, $\mathbf{Set}_*$, the category of pointed sets, is (unsurprisingly) a pointed category. It is easy to find a non-monomorphism in $\mathbf{Set}_*$ whose kernel is zero: for instance, the map $\{ 0, 1, 2 \} \to \{ 0, 1 \}$ that sends $0$ to $0$ and both $1$ and $2$ to $1$, with the distinguished element of both sets being $0$.

*Given monomorphisms $k$ and $k'$ with the same codomain, if $k$ factors through $k'$ and $k'$ factors through $k$, then the factoring morphisms are mutually inverse. But we must know that both are monomorphisms. Again, a counterexample can be found in $\mathbf{Set}_*$.
