# Does the product (by an object) in an abelian category ever have a right adjoint?

This is a follow-up to this question. Since an abelian category cannot be cartesian closed, clearly the hom functor is not right adjoint to the product (by an object). However, does the product (by an object) admit a right adjoint for some objects? If so, what is it? Does it exist in general? Does it only hold for finite products?

-