A preadditive category is a category whose hom-sets are Abelian groups and whose composition
of morphisms is additive in each argument. Thus, for composable morphisms,
and 
.
The zero element of each hom-set
is a zero morphism. An additive
functor preserves this addition on hom-sets (The
Stacks Project Authors 2026).
Every additive category is a preadditive category. The latter condition alone does not require a zero object or finite biproducts. For example, a unit ring defines a preadditive category with one object, with its elements as morphisms and its multiplication as composition of morphisms. The universal properties of a kernel and a cokernel are defined in a preadditive category whenever those morphisms exist.
