Let $\operatorname{Coim}f$ be the cokernel of the kernel pair of an arrow. Let $\operatorname{Im}f$ be the kernel of the cokernel pair.
An interesting property of a category is the isomorphy of the canonical arrow $\operatorname{Coim}\to \operatorname{Im}$. For a regular category this is equivalent to being balanced, so e.g holds for groups but fails for commutative rings.
Another interesting property of a pointed category is the isomorphy of the canonical arrow $\operatorname{Coker}\operatorname{ker}\to \operatorname{Ker}\operatorname{coker}$. This property has the important consequence that monomorphy is measured by the kernel.
I would like to have nice names for these properties but cannot really think of any good ones. At first I thought of calling the former one "the first isomorphism theorem", but then realized the first isomorphism is really about $\operatorname{Coker}\operatorname{ker}\cong \operatorname{Coim}$.
I did think about calling $\operatorname{Coker}\operatorname{ker}$ and $\operatorname{Ker}\operatorname{coker}$ respectively the linear regular coimage and linear regular image. Is this a good idea? Any suggestions in general?