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:

1. What is the ordinary Drinfeld centre, i.e. $\mathcal{Z}(\mathcal{M}_1 \boxtimes_{\mathcal{C}_2} \mathcal{M}_2)$?
2. 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 $\mathcal{Z}_{\mathcal{C}_1}(\mathcal{M}_1 \boxtimes_{\mathcal{C}_2} \mathcal{M}_2)$?

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:

1. 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.)
2. 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 $\mathcal{Z}_{\mathcal{C}_1}(\mathcal{M}_1 \boxtimes_{\mathcal{C}_2} \mathcal{M}_2)$?