The Bernstein center of a category and the Drinfeld center of a monoidal category are very similar notions: While the first one is defined as the natural endomorphisms of the identity functor on the respective category, the latter is the monoidal category of endo-pseudonatural transformations of the identity-2-functor on the delooping bicategory of the respective monoidal category, see https://ncatlab.org/nlab/show/Drinfeld+center.
By an easy identity of ends, the Bernstein center can also be written as $$Z(\mathcal{C})=\operatorname{End}(\operatorname{id}_\mathcal{C}) \cong \int_{c \in \mathcal{C}}\operatorname{Hom}(c,c)$$ Which becomes the only adequate way to define it when in the context of enriched categories onver a complete symmetric monoidal category $\mathcal{V}$ (since otherwise the natural transformations form just a set, not an element of $\mathcal{V}$).
Is the analogue for the Drinfield center, usind a lax end, also valid (which seems probable, following the discussion of 2-coends in https://arxiv.org/pdf/1501.02503.pdf, but I am not comfortable about the whole laxes and pseudos etc.) and, more importantly, does it coincide with the definition for enriched monoidal categories introduced in https://arxiv.org/pdf/1704.01447.pdf? I am not really aquainted enough with 2-categories myself to figure it out, so I would be very greatful for suggestions or more educated opinions on this (I was also told that there is a theory of centers of $E_k$-monoidal $(\infty,1)$-categories developed in HA, maybe this could be helpful to view the discussion in a different light? I don't know much about it myself though). Greetings and thanks in advance,
Markus Zetto