I'm not sure this is quite what you're looking for, but if A is small, you can consider the (contravariant) Yoneda embedding of A into the category of left additive functors from A to Ab. This is an exact full embedding, and the product of all representable functors is an injective cogenerator (this is nontrivial; it is not even obvious that left-exact functors form an abelian category). This is proven in Freyd's book Abelian Categories, and is a key part of his proof of the Mitchell Embedding Theorem. I don't know about any universal properties of this, but it is canonical.

I'm not sure this is quite what you're looking for, but if A is small, you can consider the (contravariant) Yoneda embedding of A into the category of left-exact functors from A to Ab. This is an exact full embedding, and the product of all representable functors is an injective cogenerator (this is nontrivial; it is not even obvious that left-exact functors form an abelian category). This is proven in Freyd's book Abelian Categories, and is a key part of his proof of the Mitchell Embedding Theorem. I don't know about any universal properties of this, but it is canonical.