No, this happens more often.  Let me write $A\odot X$ and $X^A$ for tensors and cotensors in a category enriched over $\mathcal{V}$. I think you are (or might be) asking for a (natural) isomorphism between $A\odot X$ and $X^A$.  In many algebraic situations, it is sensible to write $A\otimes X$ and $Hom(A,X)$.  Writing $DA = Hom(A,k)$ in $\mathcal V$, where $k$ is the unit object of $\mathcal V$, it is then sometimes true that $Hom(A,X)$ is isomorphic to $DA\otimes V$. It is also sometimes true that $A$ is isomorphic to $DA$.  That is true more often than it is true naturally, but if $\mathcal V$ is finite dimensional inner product spaces over the reals, for example, then there is a kind of contravariant naturality.  This is not a complete answer of course, but it should give you a way to think about examples.