Why are duals in a rigid/autonomous category unique up to unique isomorphism?

Question

I'm having trouble understanding the following statement:

"In a rigid category, duals are unique up to unique isomorphism."

It seems to me that this isomorphism is not unique.

Let me try to give a counterexample: Let $(X,Y,\epsilon: X \otimes Y \to I,\eta: I: Y \otimes X)$ be a dual pair (satisfying the snake identities).

Now, let $f:Y \to Y$ be any isomorphism other than the identity. We define $$h := \left(f^{-1} \otimes 1_X\right) \circ \eta$$ $$e := \epsilon \circ (1_X \otimes f)$$ Claim: $(X, Y, e, h)$ is a dual pair. Let's prove the snake identities (in a strict monoidal category): $$(e \otimes 1_X)\circ(1_X \otimes h) = \left((\epsilon \circ \left(1_X \otimes f\right)) \otimes 1_X) \circ \left(1_X \otimes \left(\left(f^{-1} \otimes 1_X\right) \circ \eta\right)\right)\right)\\ =(\epsilon \otimes 1_X) \circ (1_X \otimes f \otimes 1_X) \circ \left(1_X \otimes f^{-1} \otimes 1_X\right) \circ (1_X \otimes \eta)\\ =(\epsilon \otimes 1_X) \circ (1_X \otimes \eta) = 1_X$$ Similarly: $$(1_Y \otimes e) \circ (h\otimes 1_Y) = (1_Y \otimes (\epsilon \circ (1_X \otimes f))) \circ \left(\left(\left(f^{-1} \otimes 1_X\right) \circ \eta\right)\otimes 1_Y\right)\\ = (1_Y \otimes (\epsilon \circ (1_X \otimes f))) \circ \left(\left(\left(f^{-1} \otimes 1_X\right) \circ \eta\right)\otimes 1_Y\right)\\ = (1_Y \otimes \epsilon) \circ (1_Y \otimes 1_X \otimes f) \circ \left(f^{-1} \otimes 1_X \otimes 1_Y\right) \circ (\eta \otimes 1_Y)\\ = f^{-1} \circ (1_Y \otimes \epsilon) \circ (1_Y \otimes 1_X \otimes f) \circ (\eta \otimes 1_Y)\\ = f^{-1} \circ (1_Y \otimes \epsilon) \circ (\eta \otimes 1_Y) \circ f\\ = f^{-1} \circ f = 1_Y $$ I'm sorry for the lengthy formulae, it is an easy exercise when done in graphical calculus.

So if $Y$ has more than one automorphism (which will frequently be the case), we can define any number of different dualities. It seems that duals may be unique up to isomorphism, but this isomorphism itself is not unique. Am I wrong?

Answer 1

Thanks to Zhen Lin to bring me on the right track. The misunderstanding is the following:

The dual object ($Y$ in my example) is unique up to isomorphism, but this isomorphism is not unique. However, a dual is more than merely the object, it's the triple $(Y,\epsilon,\eta)$. This one is indeed unique up to unique isomorphism, and the calculation in the question shows exactly this.

So the same dual object $Y$ can be part of different duals.