Separable algebra objects and bimodules in a fusion category

In a monoidal category $\mathcal{C}$, an algebra object is an object $A$ with multiplication $m_A:A\otimes A\to A$ and unit ${\eta }_A:\mathbf{1}\to A$ satisfying associativity and unit equations with the associator and unitors of $\mathcal{C}$. It is separable if there is an $A$-bimodule morphism ${\sigma }_A:A\to A\otimes A$ with $m_A{\sigma }_A=1_A$. The composite $e_A={\sigma }_A{\eta }_A:\mathbf{1}\to A\otimes A$ is its separability element. It satisfies $m_A{e}_A={\eta }_A$ and the centrality equation $(m_A\otimes 1_A)\circ (1_A\otimes e_A)=(1_A\otimes m_A)\circ (e_A\otimes 1_A):A⟶A\otimes A,$ with associators inserted. In element notation these are ${\sum }_i{x}_i{y}_i=1_A$ and ${\sum }_{i}a{x}_i\otimes y_i={\sum }_i{x}_i\otimes y_{i}a$.

A left $A$-module is an object $M\in \mathcal{C}$ with an associative unital action $l_M:A\otimes M\to M$; a right $A$-module has an associative unital action $r_M:M\otimes A\to M$. Write ${\mathrm{Mod}}_{\mathcal{C}}^l(A)$ and ${\mathrm{Mod}}_{\mathcal{C}}^r(A)$ for their categories and module morphisms. The category of left $A$-modules is a right $\mathcal{C}$-module category by $M\odot X:=M\otimes X$, with left $A$-action obtained by associating $A\otimes (M\otimes X)$ to $(A\otimes M)\otimes X$ and applying $l_M\otimes 1_X$, and the inherited associator and unitor. An $A$-$B$ bimodule is an object $X\in \mathcal{C}$ with commuting left and right actions $A\otimes X\to X$ and $X\otimes B\to X$, each associative and unital. A bimodule morphism preserves both actions.

The regular $A$-$A$ bimodule is the object $A$ with both actions given by $m_A$.

Given an $A$-$B$ bimodule $X$ and a $B$-$D$ bimodule $Y$, a morphism $q:X\otimes Y\to Z$ is balanced over $B$ if $q\circ (r_X\otimes 1_Y)=q\circ (1_X\otimes l_Y):X\otimes B\otimes Y⟶Z\mathrm{.}$ A relative tensor product $X{\otimes }_{B}Y$, when it exists, is an object with a balanced map $q_B:X\otimes Y\to X{\otimes }_{B}Y$ universal among balanced maps. Existence is not part of this definition.