Filtered vector spaces can be additive with kernels and cokernels without being abelian

Let ${\mathcal{F}}_k$ be the category whose objects are $\mathbb{Z}$-filtered $k$-vector spaces $(V,F^{\bullet }V)$ and whose morphisms preserve the filtrations. Pointwise addition on linear maps and direct sums with $F^i(V\oplus W)=F^{i}V\oplus F^{i}W$ make ${\mathcal{F}}_k$ additive, and kernels and cokernels are computed on the underlying linear map with the induced and quotient filtrations.

Take $V=W=k$ with $F^{i}V=k$ for $i<0$ and $F^{i}V=0$ for $i\ge 0$, while $F^{i}W=k$ for $i\le 0$ and $F^{i}W=0$ for $i>0$. The identity linear map $\iota :V\to W$ preserves filtrations, has zero kernel and zero cokernel, so $\mathrm{coim}(\iota )=V$ and $\mathrm{im}(\iota )=W$. But $\iota$ is not an isomorphism in ${\mathcal{F}}_k$, because its inverse does not preserve $F^0$. Hence the canonical map $\mathrm{coim}(\iota )\to \mathrm{im}(\iota )$ is not an isomorphism, so ${\mathcal{F}}_k$ is not abelian.