image of a zero morphism

Let \(\mathcal{C}\) be a pointed category and \(0: A \rightarrow B\) the zero morphism for an object \(A,B \in \mathrm{Ob}(\mathcal{C})\). Then the image is the zero object.

\begin{align*} 0: 0 \rightarrow 0 = \mathrm{im}(0) \end{align*}

By definition, we have

\begin{align*} \mathrm{im}(0) = \mathrm{ker}(\pi: B \rightarrow \mathrm{coker}{0}) \end{align*}

where by cokernel of a zero morphism we get

\begin{align*} \mathrm{ker}(\mathrm{id}_{B} \rightarrow B) \end{align*}

by kernel and monomorphism in an abelian category