**Context.**   
The Drinfeld center of a rigid monoidal category $\mathcal{C}$ is again rigid. This is not hard to see: Given an object $X\in \mathcal{C}$ together with a half-braiding $\phi_X:X\otimes-\xrightarrow{\sim} -\otimes X$, one can endow the left (resp. right) dual object of $X$ with a half-braiding using the respective evaluations and coevaluations in $\mathcal{C}$. For details see [Theorem 2.23 in this PhD thesis](https://inspirehep.net/files/6fd8c960abc84d4a12a08897031ef0f9#page=47), for instance. 

I am interested in the Drinfeld center of non-rigid closed monoidal categories. 
For $R$ a commutative ring, the Drinfeld center of the symmetric monoidal category $\operatorname{Mod}(R)$ of (left) $R$-modules is monoidally equivalent to $\operatorname{Mod}(R)$ itself; see [here](https://mathoverflow.net/questions/385792/drinfeld-center-of-mathrmmod-r). In particular, it is closed monoidal.

**Question.**  
Have Drinfeld centers of other non-rigid closed monoidal categories been considered in the literature? If so, where? 

Ideally, someone has described the Drinfeld centers of a few naturally ocurring non-rigid closed monoidal categories by showing that they are each monoidally equivalent to another known monoidal category. Such a stock of examples would be useful to me in testing some ideas and conjectures.