Variant 6.1.6.7. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $\operatorname{coev}: {\bf 1} \rightarrow Y \otimes X$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the composite map
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes C, D) & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y \otimes (X \otimes C), Y \otimes D) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (Y \otimes X) \otimes C, Y \otimes D) \\ & \xrightarrow {\operatorname{coev}} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( {\bf 1} \otimes C, Y \otimes D) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Y \otimes D ) \end{eqnarray*}is a bijection.
For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the composite map
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes Y, D) & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (C \otimes Y) \otimes X, D \otimes X ) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes (Y \otimes X), D \otimes X) \\ & \xrightarrow {\operatorname{coev}} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C \otimes {\bf 1}, D \otimes X ) \\ & \simeq & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, D \otimes X ) \end{eqnarray*}is a bijection.
There exists a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ for which the pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum, in the sense of Definition 6.1.6.1.
Moreover, if these conditions are satisfied, then the morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ is unique.