# How do the first three axioms of an abelian category imply that hom-sets are enriched over the monoidal category of abelian groups? [duplicate]

According to this article in Wikipedia the following first three axioms in the definition of of an abelian category imply that hom-sets are enriched over the monoidal category of Abelian groups:

• existence of a zero object
• existence of binary products and binary coproducts
• existence of all kernels and cokernels

(see this image for a saved copy of the relevant section)

My intuition is that it should follow from a diagram like this:

$$A\to A\otimes A \to B \oplus B \to B$$

where the first map is the diagonal map, the second is the product of the two maps one wishes to sum, i.e. $\phi\times \psi$ and the third map is the sum. But how could I use the first and third axioms to prove the existence of the diagonal and sum maps?

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## marked as duplicate by Zhen Lin, Did, Yiorgos S. Smyrlis, Claude Leibovici, user127.0.0.1Feb 23 '14 at 18:35

$id_B : B \to B$ and $id_B : B \to B$ give $(id_B,id_B) : B \oplus B \to B$ (univ.prop. of coproduct). – Martin Brandenburg Feb 23 '14 at 21:53