Let $C$ be a symmetric monoidal category.

• Recall that a dual for $X \in C$ is an object $X^\vee$ and maps $\eta: I \to X \otimes X^\vee$ and $\varepsilon: X^\vee \otimes X \to I$ (where $I$ is the monoidal unit) satisfying the triangle identities.

• Let's say that $X$ is self-dual if there is a dual $X^\vee$ for $X$ with $X \cong X^\vee$.

• Let's say that $X$ is idempotent if $X \cong X \otimes X$.

• Let's say that $X$ is well-idempotent if there is a map $i: I \to X$ such that $X \otimes i: X \to X \otimes X$ is an isomorphism.

• Dually, $X$ is co-well-idempotent if there is a map $p: X \to I$ such that $X \otimes p: X \otimes X \to X$ is an isomorphism.

Clearly, if $X$ is self-dual and idempotent, then $X$ is both well-idempotent and co-well-idempotent. Conversely, I ask

Questions:

1. If $X$ is idempotent and dualizable, then is $X$ self-dual?

2. If $X$ is well-idempotent and dualizable, then is $X$ self-dual?

3. If $X$ is well-idempotent and co-well-idempotent and dualizable, then is $X$ self-dual?

I suspect the answer to all these questions is "no", but I don't know any examples.