Pointed category

A pointed category has a zero object, hence a distinguished zero morphism between every pair of objects.

A zero object is both an initial object and a terminal object. In a pointed category, the composite through the zero object is the zero morphism $0_{A,B}:A\to B$.

The kernel of $f:A\to B$ is the universal morphism $k:K\to A$ satisfying $fk=0$. It is the equalizer of $f$ and the zero morphism.

The cokernel of $f:A\to B$ is the universal morphism $q:B\to Q$ satisfying $qf=0$. It is the coequalizer of $f$ and the zero morphism.

A normal monomorphism is a monomorphism that is the kernel in a category of some morphism. In a pointed category with kernels and cokernels, a monomorphism is normal exactly when it is the kernel of its own cokernel.

A conormal epimorphism is an epimorphism that is the cokernel in a category of some morphism. It is the concept dual to a normal monomorphism.