The following is a straightforward generalization of the notion of dualizable object in a symmetric monoidal category given in Duality, Trace and Transfer by Albrecht Dold and Dieter Puppe to non-symmetric monoidal categories. Throughout this question $(\mathcal{C}, \otimes, I)$ denotes a monoidal category.
Definition: Let $X$ be an object in $\mathcal{C}$. A right dual of $X$, if it exists, is a pair $(X',\, \varepsilon: X' \otimes X \to I)$ such that for every object $Y \in \mathcal{C}$ the map $$ \begin{array}{rcl} \mathcal{C}(Y,X') & \to & \mathcal{C}(Y \otimes X, I) \\ f & \mapsto & \varepsilon \circ (X \otimes f) \end{array} $$ is a bijection. If such a right dual exists we say that $X$ is weakly right dualizable. The notion of a left dual and weakly left dualizablity is defined... dually. If $\mathcal{C}$ is symmetric monoidal, then the two notions coincide, and we simply speak of the dual of $X$, and say that $X$ is weakly dualizable.
Call $\mathcal{C}$ left-closed (resp. right-closed) if for every object $Y \in \mathcal{C}$ the functor $-\otimes Y$ (resp. $Y \otimes -$) admits a left-adjoint $F_l$ (resp. $F_r$), then the left dual (resp. right dual) of $X$ is given by $F_l(X,I)$ (resp. $F_r(X,I)$). If $\mathcal{C}$ is symmetric monoidal, then $F: = F_l = F_r$, and the dual of $X$ is given by $F(X,I)$. In this way we obtain many examples of weakly dualizable objects, as any object in closed symmetric monoidal is weakly dualizable, so e.g. every modules over a ring is weakly dualizable.
Question: What are examples of weakly dualizable objects in non-closed monoidal categories. I would be particularly curious about examples of weakly dualizable objects in such categories that are non-symmetric.