In the category of modules, the tensor product functor is the left adjoint of the covariant Hom functor. Similarly in the category of sets, the Cartesian product functor is the left adjoint of the covariant Hom functor. I’m wondering whether this can be generalized.
Let $C$ be a category where the covariant Hom functor has a left adjoint. My question is, does this left adjoint make $C$ into a monoidal category? That is, does it induce a tensor product on $C$, in the sense that it is the tensor product functor of some tensor product on $C$?
If not, does anyone know of a counterexample?