i.e. we have 0fR----------->S| | | | |0_{RU}|0_{SV}| | VgVU----------->V

By taking *f* or *g* to be the identity morphism in the diagram above, we see that the composition of any morphism with a zero morphism results in a zero morphism. Furthermore, if a category has a family of zero morphisms, then this family is unique.

If a category has zero morphisms, then one can define the notions of kernel and cokernel in that category.

- The zero morphisms in the category of groups or modules as introduced above are zero morphisms in the new general sense.
- If
**C**is a preadditive category, then every morphism set Mor(*X*,*Y*) is an abelian group and therefore has a zero element. These zero elements form a family of zero morphisms for**C**. - If
**C**has a zero object*Z*, then from*X*there is a unique morphism to*Z*, and from*Z*there is a unique morphism to*Y*. Composing these two gives a morphism from*X*to*Y*. The family of all morphisms so constructed is a family of zero morphisms for**C**. - The category of all sets with functions as morphisms does not have zero morphisms; neither does the category of all topological spaces, with continuous maps as morphisms.