Definition 6.1.6.1. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$. A duality datum is a pair $(\operatorname{coev}, \operatorname{ev})$, where $\operatorname{coev}: {\bf 1} \rightarrow Y \otimes X$ and $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ are morphisms of $\operatorname{\mathcal{C}}$ satisfying the following compatibility conditions:
- $(Z1)$
The composition
\[ X \xrightarrow [\sim ]{\rho _{X}^{-1}} X \otimes {\bf 1} \xrightarrow {\operatorname{id}_ X \otimes \operatorname{coev}} X \otimes (Y \otimes X) \xrightarrow [\sim ]{\alpha _{X,Y,X}} (X \otimes Y) \otimes X \xrightarrow {\operatorname{ev}\otimes \operatorname{id}_ X} {\bf 1} \otimes X \xrightarrow [\sim ]{\lambda _ X} X \]is equal to the identity morphism $\operatorname{id}_{X}$. Here the isomorphism $\alpha _{X,Y,X}$ is the associativity constraint for the monoidal category $\operatorname{\mathcal{C}}$, and the isomorphisms $\lambda _{X}$ and $\rho _{X}$ are the left and right unit constraints of Construction 2.1.2.17.
- $(Z2)$
The composition
\[ Y \xrightarrow [\sim ]{\lambda _{Y}^{-1} } {\bf 1} \otimes Y \xrightarrow {\operatorname{coev}\otimes \operatorname{id}_ Y} (Y \otimes X) \otimes Y \xrightarrow [\sim ]{\alpha _{Y,X,Y}^{-1}} Y \otimes (X \otimes Y) \xrightarrow {\operatorname{id}_{Y} \otimes \operatorname{ev}} Y \otimes {\bf 1} \xrightarrow [\sim ]{\rho _ Y} Y \]is equal to the identity morphism $\operatorname{id}_{Y}$.
If these conditions are satisfied, then we will refer to $\operatorname{coev}$ as the coevaluation morphism of the duality datum $(\operatorname{coev}, \operatorname{ev})$, and to $\operatorname{ev}$ as the evaluation morphism of the duality datum $(\operatorname{coev}, \operatorname{ev})$. In this case, we say that the pair $(\operatorname{coev}, \operatorname{ev})$ exhibits $X$ as a left dual of $Y$, also that it exhibits $Y$ as a right dual of $X$.