Implication details

Proof: Choose an object $A$ and a kernel $k : K \to A$ of $\id_A$. Then $k = \id_A \circ k = 0_{A,A} \circ k = 0_{K,A}$. For every object $T$, the zero morphism $0_{T,K}$ provides a morphism $T \to K$. Conversely, if $f : T \to K$ is any morphism, then $k \circ f = k \circ 0_{T,K}$ since $k$ is a zero morphism, hence $f = 0_{T,K}$, using that $k$ is also a monomorphism. Thus, $K$ is a terminal object.Remark: With this result it follows that $K$ is even a zero object.