Lemma 4.43.10. Let $(\mathcal{C}, \otimes , \phi , \psi )$ be a symmetric monoidal category. Let $X$ be an object of $\mathcal{C}$ and let $Y$, $\eta : \mathbf{1} \to X \otimes Y$, and $\epsilon : Y \otimes X \to \mathbf{1}$ be a left dual of $X$ as in Definition 4.43.5. Then $\eta ' = \psi \circ \eta : \mathbf{1} \to Y \otimes X$ and $\epsilon ' = \epsilon \circ \psi : X \otimes Y \to \mathbf{1}$ makes $X$ into a left dual of $Y$.

**Proof.**
Omitted. Hint: pleasant exercise in the definitions.
$\square$

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