Warning 6.1.6.16. In the situation of Proposition 6.1.6.15, it is possible for an object $X \in \operatorname{\mathcal{C}}$ to admit a weak right dual which is not a right dual. For example, let $\operatorname{\mathcal{C}}= \operatorname{Vect}_{k}$ be the category of vector spaces over a field $k$, equipped with the monoidal structure of Example 2.1.3.1. Let $V$ be a vector space over $k$ and let $V^{\ast } = \operatorname{Hom}_{k}(V, k)$ be its dual space. Then the evaluation map
\[ \operatorname{ev}: V \otimes _{k} V^{\ast } \rightarrow k \quad \quad v \otimes \lambda \mapsto \lambda (v) \]
exhibits $V^{\ast }$ as a weak (right) dual of $V$ (in the sense of Definition 6.1.6.13). However, it is a duality datum only when $V$ is finite-dimensional over $k$ (Exercise 6.1.6.12).