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

$\vcenter { \xymatrix{ X \ar[rd]_1 \ar[r]_-{\eta \otimes 1} & X \otimes Y \otimes X \ar[d]^{1 \otimes \epsilon } \\ & X } } \quad \text{and}\quad \vcenter { \xymatrix{ Y \ar[rd]_1 \ar[r]_-{1 \otimes \eta } & Y \otimes X \otimes Y \ar[d]^{\epsilon \otimes 1} \\ & Y } }$

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

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