A zero morphism 
in a category is a morphism
satisfying 
for all morphisms 
and 
for all morphisms 
, for arbitrary objects 
and 
(Pareigis 1970, pp. 22-23).
A category with zero morphisms has a distinguished morphism 
for every two objects 
and 
. For morphisms 
and 
, 
and 
. These identities make each
a zero morphism, and the family
satisfying them is unique (Pareigis 1970, p. 23).
If a zero object 
exists, 
is the composition
of morphisms 
of the unique morphisms to
and from 
. A family of zero morphisms can exist without a zero object. Every preadditive
category has such a family, given by the identity
elements of its Abelian group hom-sets.
In the category of Abelian
groups, these are the zero maps (Pareigis 1970, pp. 23-24;
The Stacks Project Authors 2026).
