TITLE: Contravariant internal hom
QUESTION [6 upvotes]: 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.

REPLY [8 votes]: Suppose that everything is dualizable (as in the category of finite-dimensional vector spaces over a field, for example), and write $TX=[X,A]=X^*\otimes A$.  Then 
\begin{align*}
 \mathcal{C}(TX,Y) &= 
  \mathcal{C}(1,X\otimes Y\otimes A^*) =
  \mathcal{C}(Y^*\otimes A,X) = 
   \mathcal{C}^{\text{op}}(X,TY) \\
 \mathcal{C}(W,TX) &= 
  \mathcal{C}(1,W^*\otimes X^*\otimes A) =
  \mathcal{C}(X,W^*\otimes A) = 
   \mathcal{C}^{\text{op}}(TW,X)
\end{align*}
so $T$ is self-adjoint on both sides.