additive functor and biproduct
1. Proposition
Let 
be additive categories.
TFAE:
is an additive functor- preserves finite biproducts (including the zero object)
2. Proof
2.1. 1) 
2)
2.1.1. zero object
Let 
be the zero object, then
Then by functorality, it follows, that
Furthermore it follows, that
is a group-homomorphism, thus
Therefore since 
we conclude that
thus it follows by zero object and zero morphism as identity $(0) = 0$d
2.1.2. finite product
follows from biproduct in an ab-enriched category determined by morphisms
as 
preserves zero morphisms as stated above and 
as functor, i.e.
and furthermore
