ct.category theory - Does the product (by an object) in an abelian category ever have a right adjoint?

In an additive category the functor F(–) = – × A = – ⊕ A cannot have a right adjoint unless A = 0. If F had a right adjoint then it would preserve coproducts and in particular A = F(0) = F(0 ⊕ 0) = F(0) ⊕ F(0) = A ⊕ A via the fold map. This means Hom(A, K) = Hom(A, K) × Hom(A, K) for every K, but Hom(A, K) is nonempty (we have zero maps) so Hom(A, K) = • and thus A = 0 by Yoneda.