I have troubles proving the following statement although I verified it for a large class of finite tensor categories.

Let $ \mathcal C$ be a finite tensor category and $ \mathcal D$ be a tensor subcategory of $ \mathcal C$ . One can consider the central Hopf monad

$ $ Z(V)=\int_{X\in \mathcal C} X\otimes V \otimes X^*.$ $

Let $ \pi_{V, X}: Z(V)\rightarrow X\otimes V \otimes X^*$ be the canonical maps associated to the coend $ Z(V)$ for all $ X\in \mathcal C$ .

It is well known that $ Z$ is a Hopf comonad and it has values in the Drinfeld center $ \mathcal Z(\mathcal C)$ . Thus each object $ Z(V)$ can be enhanced with a half-braiding $ (Z(V), \sigma_{V, -})\in \mathcal Z(\mathcal C)$ .

Moreover $ $ R:\mathcal C \rightarrow \mathcal Z(\mathcal C),\;\; V \mapsto (Z(V), \sigma_{V,-})$ $ is a right adjoint of the forgetful functor $ F: \mathcal Z(\mathcal C)\rightarrow \mathcal C$ , see the paper of Day and Street, Centres of monoidal categories of functors, in: Categories in Algebra, Geometry and Mathematical Physics, in: Contemp. Math., vol.431, Amer. Math. Soc., Providence, RI, 2007, pp.187–202. We also have $ Z=FR$ .

One can also consider the relative Hopf comonad

$ $ U(V)=\int_{X\in \mathcal D} X\otimes V \otimes X^*.$ $

Let also $ \bar{\pi}_{V, X}: U(V)\rightarrow X\otimes V \otimes X^*$ be the canonical maps associated to the coend $ U(V)$ for all $ X\in \mathcal D$ .

Similarly, one can see that $ U(V)$ can be enhanced with a relative braiding $ \bar{\sigma}_{U(V), -}$ . Thus $ (U(V), \bar{\sigma}_{U(V), -})\in \mathcal Z_{\mathcal C}(\mathcal D)$ , the relative center with respect to $ \mathcal D$ . Thus $ $ R_1: \mathcal C \rightarrow Z_{\mathcal C}(\mathcal D), \;\; V\mapsto (U(V), \bar{\sigma}_{U(V), -}) )$ $ is a right adjoint of the forgetful functor $ F_1:Z_{\mathcal C}(\mathcal D) \rightarrow \mathcal C$ and $ U=R_1F_1$ .

By the universal property of the coend $ U$ , for any $ V\in \mathcal C$ , there is a natural transformation $ $ q_V:Z(V)\rightarrow U(V)$ $ such that $ \bar{\pi}_{V, X}\circ q_V=\pi_{V, X}$ for all $ X\in \mathcal D$ . This natural transformation also appear in the paper of Shimizu, The monoidal center and the character algebra, JPAA 221 (2017) 2338–2371, see https://arxiv.org/abs/1504.01178.

One can also consider the following sequence of functors:

$ $ \mathcal Z(\mathcal C)\xrightarrow{F_2} {\mathcal Z}_{\mathcal C} (\mathcal D) \xrightarrow{F_1} \mathcal C$ $

where the first category is as above the Drinfeld center of $ \mathcal C$ and the second category is the relative center with respect to $ \mathcal D$ .

The two functors have right adjoints denoted by $ R_2$ and respectively $ R_2$ . The functor $ R_1$ was described above.

On the other hand, we have $ F=F_1F_2$ and therefore we may assume $ R=R_2R_1$ .

With these notations, it can be seen that the unit $ \eta^2:\mathrm{id}_{{\mathcal Z}_{\mathcal C} (\mathcal D)} \rightarrow F_2R_2$ of the adjunction $ (F_2,R_2)$ induces a natural transformation

$ m: Z(V)=FR(V)\rightarrow U(V)=F_1R_1(V)$ .

**My question is if $ m=q$ as natural transformations?**

**Note:** It can be relatively easy to check that this happens in the case $ \mathcal C=\mathrm{rep}(H)$ , the category of representations of a finite dimensional Hopf algebra.