Let $\mathcal{C}$ be a braided $\mathbb{k}$-linear fusion category ($\mathbb{k}$ algebraically closed; if necessary to answer my question you can also assume $\mathcal{C}$ to be pivotal or even modular). Two algebras $A,B$ in $\mathcal{C}$ are called *Morita equivalent* if the categories $A\hbox{-}\mathsf{Mod}(\mathcal{C})$ and $B\hbox{-}\mathsf{Mod}(\mathcal{C})$ of (left or right) modules internal to $\mathcal{C}$ are equivalent.

**Questions:**

(1) Is there any alternative, equivalent characterization of Morita equivalence of two algebras which circumvents showing that their module categories are equivalent?

(2) In particular, is there a way to "calculate" the Morita equivalence class of a given algebra?

(3) Is there anything special one can say about the Morita equivalence class of an algebra, e.g. some classification result, maybe by imposing additional properties on the algebra?

Please also refer to literature, thanks!