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

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.

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