Let $C$ be a symmetric monoidal category. I am interested in objects $X \in C$ such that the symmetry

$S_{X,X} : X \otimes X \cong X \otimes X$

is equal to the identity. There are many examples of such objects, e.g. invertible sheaves. My first question is: How would you call such an object?

Now assume that $X$ has a dual $Y$, i.e. we have morphisms $e: Y \otimes X \to 1$ and $c : 1 \to X \otimes Y$ such that the triangular identities are satisfied.

Question. Assuming $S_{X,X}$ is the identity, can we conclude that $S_{Y,Y}$ is the identity? If not, does it suffice to assume that $e$ (and thus $c$) is an isomorphism?