If $A$ and $B$ are dualizable objects in a monoidal category, is the unit of the one duality the inverse of the counit of the other duality?

Question

I'm currently trying to wrap my head around dualizable objects in monoidal categories and I was wondering whether the following claim holds:

Let $A$ and $B$ be dualizable objects in a monoidal category, i.e. $A \dashv B$ with unit $\eta : I \to B \otimes A$ and $B \dashv A$ with counit $\varepsilon' : B \otimes A \to I$. Then $\eta \circ \varepsilon' = \text{id}_{B \otimes A}$.

I've been trying to manipulate the diagrams (triangle identities) in this definition to convince myself of my claim, but to no avail. I'm getting the suspicion that my claim is actually false. What am I missing?

Answer 1

Consider a vector space $V$ over a field $k$ of finite dimension $n>1$. Then you are asking if a certain composite $V^*\otimes V\to k\to V^*\otimes V$ is the identity map. This cannot hold for reasons of dimension, so the claim is generally false.