I am working with the following definition of abelian category.
a) It has a $0$ object.
b) Every morphism has a kernel and a cokernel. Every monomorphismis a kernel and every epimorphism is a cokernel.
c) It has all finite products and finite coproducts.
Now, my definition does not make it explicit that finite products and coproducts are the same, but I know that that holds for all abelian categories.
More precisely, I would like to prove that for all objects $A$ and $B$, the morphism $ (id_A, id_B) : A \bigoplus B \longrightarrow A \times B$ is iso. Note that the morphism I've written can be constructed by using the universal properties of product and coproduct.