Let $A$ be an additive category. Namely
- $A$ has a zero object,
- $A$ has finite products and coproducts, and
- Every Hom-set is an Abelian group such that composition of morphisms is bilinear.
Question: Is the zero morphism always the additive unit?
Since $A$ has a zero object, say $0$, for any two objects $c$ and $d$, there uniquely exists a morphism $c \to d$ called the zero morphism which can be decomposed as $c \to 0 \to d$.
For typical examples (e.g. (Ab), ($R$-Mod), (Chain complexes in $R$-Mod)), the zero map gives the additive unit of each Hom-set. But the axioms of additive categories do not seem to say that the zero maps give the additive units of Hom-sets.
Is it possible to show that the zero morphism is the additive unit of each Hom-set? Or are there any additive categories such that the zero morphism is not the additive unit of some Hom-set? How about abelian categories?