Let $ \mathcal{C}$ be a fusion category, and $ \mathcal{M}, \mathcal{N}$ semisimple $ (\mathcal{C}, \mathcal{C})$ -bimodule categories. The left $ \mathcal{C}$ -action is denoted as $ – \triangleright – \colon \mathcal{C} \boxtimes \mathcal{M} \to \mathcal{M}$ , analogously the right action.

- A
**relative half-braiding**on an object $ M \in \operatorname{ob} \mathcal{M}$ is an natural isomorphism $ \gamma_C \colon C \triangleright M \to M \triangleleft C$ , satisfying the hexagon equation. - The category of objects in $ \mathcal{M}$ equipped with relative half-braidings, and compatible morphisms, is called the
**relative center**, denoted as $ \mathcal{Z}_{\mathcal{C}}(\mathcal{M})$ . It is a $ \mathcal{Z}(\mathcal{C})$ -bimodule category, where $ \mathcal{Z}(\mathcal{C}) = \mathcal{Z}_\mathcal{C}(\mathcal{C})$ is the conventional Drinfeld center. - A
**balanced functor**$ F\colon \mathcal{M}_1 \boxtimes \mathcal{M}_2 \to \mathcal{N}$ is a functor equipped with a natural isomorphism $ \beta_{M_1,C,M_2}\colon F(M_1 \triangleleft C \boxtimes M_2) \to F(M_1 \boxtimes C \triangleright M_2)$ satisfying certain axioms. - The
**relative Deligne product**, denoted $ \mathcal{M}_1 \boxtimes_\mathcal{C} \mathcal{M}_2$ , is the universal category with a balanced functor $ \mathcal{M}_1 \boxtimes \mathcal{M}_2 \to \mathcal{M}_1 \boxtimes_\mathcal{C} \mathcal{M}_2$ . Intuitively, one thinks of it as $ \mathcal{M}_1 \boxtimes \mathcal{M}_2$ “modulo” the $ \mathcal{C}$ -action.

**Fact**: $ \mathcal{Z}_\mathcal{C}(\mathcal{M}_1 \boxtimes_\mathcal{C} \mathcal{M}_2) \simeq \mathcal{Z}_\mathcal{C}(\mathcal{M}_1) \boxtimes_{\mathcal{Z}(\mathcal{C})} \mathcal{Z}_\mathcal{C}(\mathcal{M}_2)$

Proof: See e.g. Fusion categories and homotopy theory (Pavel Etingof, Dmitri Nikshych, Victor Ostrik), Proposition 3.11.

**Question**: Is there a generalisation of this formula? There are at least two other cases where a similar formula might be expected:

- Let $ \mathcal{M}_i$ be monoidal. What is the ordinary Drinfeld centre, i.e. $ \mathcal{Z}(\mathcal{M}_1 \boxtimes_{\mathcal{C}} \mathcal{M}_2)$ ? (If it helps, you may assume that $ \mathcal{C}$ is braided and its action factors through central functors.)
- Assume that $ \mathcal{M}_1$ is a $ (\mathcal{C}_1, \mathcal{C}_2)$ -bimodule category, and $ \mathcal{M}_2$ is a $ (\mathcal{C}_2, \mathcal{C}_1)$ -bimodule category. What can we say about $ \mathcal{Z}_{\mathcal{C}_1}(\mathcal{M}_1 \boxtimes_{\mathcal{C}_2} \mathcal{M}_2)$ ?