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).
