Show that $A\to B$ is a kernel of $B\to C$ in a sequence $0\to A\to B\to C$.

Question

Assume that in a category $\mathcal{C}$ the sequence $0\to A\to B\to C$ is cokernel-exact and kernel-exact and $\mathcal{C}$ is balanced. Show that $A\to B$ is a kernel of $B\to C$. You can assume, if needed, that $\mathcal{C}$ is pre-abelian.

Thanks for any help.