Let $\mathcal{C}$ a symmetric monoidal category. By using symmetry, it is very easy to show that the contravariant internal hom functor $[-,A]\colon\mathcal{C}^{op}\longrightarrow\mathcal{C}$ is adjoint on the right to itself.

My question is: when is this functor additionally $[-,A]$ a left-adjoint? Do such objects $A$ have a name? In the category of $\textbf{Set}$ it is easily seen that $[-,A]$ preserves the terminal object $\textbf{Set}^{op}$ iff $A$ is the empty set.