For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{\*})^\*$, for $\*$ denoting dual? If so, what is a good example.



The dual of a dual in a rigid tensor category
As Tobias said in his answer, a good place to look for examples is in endofunctor categories with composition as the monoidal product, where duals are adjoints. But another way to get a rigid monoidal category of such is to just restrict to the full subcategory of endofunctors $F = F\_0$ that fit into some infinite adjoint string


$$ \cdots \dashv F\_{-2} \dashv F\_{-1} \dashv F\_0 \dashv F\_1 \dashv F\_2 \dashv \cdots$$


Since composition preserves adjoints, this subcategory is closed under the monoidal structure, hence is itself a monoidal category; and it is evidently rigid since each of the $F\_n$ above also belongs to it (just shift the adjoint string back and forth).


So now we just need to find an endofunctor that fits into such an infinite adjoint string but whose left and right adjoints are not isomorphic. One such "naturally ocurring" example appears in Remark 6.11 of [arXiv:1704.08084](https://arxiv.org/abs/1704.08084): if $D$ is a stable derivator, then there is an infinite adjoint string


$$ \cdots \dashv G\_{-2} \dashv G\_{-1} \dashv G\_0 \dashv G\_1 \dashv G\_2 \dashv \cdots$$


connecting $D$ and $D^{[1]}$, where $[1]$ is the arrow category $(0\to 1)$. These are not endofunctors, but composing them in pairs we get an adjoint string of endofunctors:


$$ \cdots \dashv G\_{-1} G\_{-2} \dashv G\_{-1} G\_0 \dashv G\_1 G\_0 \dashv G\_1 G\_2 \dashv G\_3 G\_2 \dashv \cdots$$


Now $G\_1: D \to D^{[1]}$ is the constant diagram functor, so that $G\_2$ is evaluation at $0$ and $G\_{0}$ is evaluation at $1$. Then $G\_3$ sends an object $X$ to the morphism $X\to 1$, and $G\_{-1}$ sends it to the morphism $0\to X$. Thus $G\_1 G\_0(X\to Y) = (Y\to Y)$, while $G\_1 G\_2(X\to Y) = (X\to X)$ and $G\_{-1} G\_0(X\to Y) = (0\to Y)$. So as long as $D$ is nontrivial, $G\_{-1} G\_0$ is an object in a rigid monoidal category that is not isomorphic to its double right dual.


