Relative Deligne product in the braided module 2-category

Let $\mathcal{C}$ be a braided fusion category and let $\mathcal{M},\mathcal{N}$ be finite semisimple right $\mathcal{C}$-module categories. A relative Deligne product $\mathcal{M}{⊠}_{\mathcal{C}}\mathcal{N}$ in $\mathrm{Mod}(\mathcal{C})$ means a finite semisimple right $\mathcal{C}$-module category $\mathcal{P}$ equipped with an exact bilinear functor $b:\mathcal{M}\times \mathcal{N}\to \mathcal{P}$ that is balanced over the middle $\mathcal{C}$-action and compatible with the outer right $\mathcal{C}$-action, and is universal among such functors: for every finite semisimple right $\mathcal{C}$-module category $\mathcal{D}$, composition with $b$ gives an equivalence between right $\mathcal{C}$-module functors $\mathcal{P}\to \mathcal{D}$ and exact bilinear middle-balanced right $\mathcal{C}$-module functors $\mathcal{M}\times \mathcal{N}\to \mathcal{D}$.

For the balancing convention, a right module category $\mathcal{N}$ is regarded as a left module category by $X▹n:=n\odot X$. Its left module associator $(X\otimes Y)▹n\to X▹(Y▹n)$ is the composite of $n\odot c_{Y,X}^{-1}$ with the inverse right-module associator. “Balanced” means that $b(m\odot X,n)\cong b(m,X▹n)$ naturally in all variables, compatibly with the unit and associativity constraints. This definition fixes the universal property and the braiding convention; it does not assert that a product exists.