Under which conditions can we say that the cotensor objects in a (closed) V-category are the exponential objects? It is just when V=Set?



Cotensor vs exponential objects.
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 $DA\otimes X$. 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.


