[2] The zero object of Ab is the trivial group {0} which consists only of its neutral element.
This addition of morphism turns Ab into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.
The same is true for cokernels; the cokernel of f is the quotient group C = B / f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp: in Grp it can happen that f(A) is not a normal subgroup of B, and that therefore the quotient group B / f(A) cannot be formed.)
With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an abelian category.
With this notion of product, Ab is a closed symmetric monoidal category.