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Given a monoidal category ${\cal C}$, and $X \in {\cal C}$, we define a left dual of $X$ to be an object $X^*$ such that there exist morphisms $\epsilon:X^* \otimes X \to I$, and $\eta:I \to X \otimes X^*$, for $I$ the identity of the category, satisfying certain axioms, see here for details.

When is the left of a dual unique up to isomorphism, and when is this isomorphism unique.

Najib Idrissi
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Christian Fischmann
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2 Answers2

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They are always isomorphic, and by a unique isomorphism that preserves the structure of the duality (see this question for the unicity).

Suppose $X^*$ and $\hat{X}^*$ are two different left duals (with corresponding evaluations $\epsilon, \hat\epsilon$ and coevaluations $\eta, \hat\eta$). Define $f : \hat{X}^* \to X^*$ as the composite $$f = (\hat{\epsilon} \otimes 1) \circ (1 \otimes \eta) : \hat{X}^* \xrightarrow{1 \otimes \eta} \hat{X}^* \otimes X \otimes X^* \xrightarrow{\hat\epsilon \otimes 1} X^*$$ Similarly define $\hat{f} = (\epsilon \otimes 1) \circ (1 \otimes \hat{\eta})$. (In both cases I left out the isomorphisms $I \otimes Y \cong Y \cong Y \otimes I$ and the associators, but if you wanted to be 100% precise you should include them). Then it is a tedious but completely mechanical exercise to check that $f$ and $\hat{f}$ are inverse to each other, because of the two axioms (applied respectively to $X^*$ and $\hat{X}^*$) in the Wikipedia article you cited.

Community
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Najib Idrissi
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  • Can you expand on the mechanical details? If one computes $f\circ\hat{f}$, it expands to $\hat{X}^\rightarrow \hat{X}^\otimes X\otimes X^* \rightarrow X^* \rightarrow X^\otimes X\otimes \hat{X}^ \rightarrow \hat{X}^*$, but there's nowhere in there that I can see that corresponds to the morphisms in the definition of the dual. – Sam Jaques Aug 11 '19 at 01:50
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This is much less clear than claimed by the previous post. See this stackexchange post for the full proof.

Kofi
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