Let $\mathcal{C}$ be a category. Then $\mathcal{C}$ is an additive category if

1. $\mathcal{C}$ is a preadditive category, and
2. for every pair of objects $A,B$ in $\mathcal{C}$ , their product exists.

Proposition. In a preadditive category, coproduct of two objects exists iff their product exists. Furthermore, they are isomorphic.

Proof. We shall prove the fact if the product $D$ of objects $A$ and $B$ exists, then $D$ is also their coproduct. The other direction is dual.

Suppose $D$ is the product of $A$ and $B$ , with morphisms

    and

From these two morphisms, we construct two commutative diagrams

where $0$ and $1$ are zero morphisms and identity morphisms on $A$ and $B$ , and $\alpha$ and $\beta$ are morphisms based on the definition of the product $D$ .

Then it's not hard to see that $D$ is a coproduct of $A$ and $B$ with morphisms $\alpha$ and $\beta$ , for if $r:A\rightarrow C$ and $s:B\rightarrow C$ are two morphisms into an object $C$ , we can form two morphisms $r\pi_A$ and $s\pi_B$ , both from $D$ to $C$ . Since $\operatorname{hom}(D,C)$ is an abelian group, these two can then be added to form $f:=r\pi_A+s\pi_B$ . Then $f\alpha=(r\pi_A+s\pi_B)\alpha=r$ , and similarly $f\beta=s$ . This shows that $D$ is also the coproduct of $A$ and $B$ with morphisms $\alpha$ and $\beta$ .

As a result of the above proposition, in an additive category, finite products and finite coproducts are synonymous. Given objects $A,B$ , we denote $A\oplus B$ to be their product. We also call it the direct sum of $A$ and $B$ .

Many preadditive categories are also examples of additive categories. The category ${CyclGrp}$ of cyclic groups as the subcategory of the category of abelian groups is an example of a preadditive category that is not additive, for the product of two cyclic groups $\mathbb{Z}/p \mathbb{Z}$ and $\mathbb{Z}/q \mathbb{Z}$ exists in ${CyclGrp}$ only when $p$ and $q$ are coprime.