Definition 4.43.5. Given a monoidal category $(\mathcal{C}, \otimes , \phi )$ and an object $X$ a *left dual* is an object $Y$ together with morphisms $\eta : \mathbf{1} \to X \otimes Y$ and $\epsilon : Y \otimes X \to \mathbf{1}$ such that the diagrams

commute. In this situation we say that $X$ is a *right dual* of $Y$.

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