Gluing vertex algebras

We relate commutative algebras in braided tensor categories to braid-reversed equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a category, we give detailed construction of the canonical $\mathcal{C}\boxtimes\mathcal{C}^\text{rev}$: if is semisimple but not necessarily finite or rigid, then $\bigoplus_{X\in\text{Irr}(\mathcal{C})}X'\boxtimes X$ algebra, with $X'$ representing object $\text{Hom}_\mathcal{C}(\bullet\otimes_\mathcal{C}X,\mathbf{1}_{\mathcal{C}})$. Conversely, let $A=\bigoplus_{i\in I}U_i\boxtimes V_i$ be simple $\mathcal{U}\boxtimes\mathcal{V}$ $\mathcal{U}$ and rigid finite, $\mathcal{V}$ semisimple. If unit objects form commuting pair $A$, show there equivalence between subcategories sending $U_i$ $V_i^*$. When are module operator $U$ $V$, glue $V$ along via map $\tau:\text{Irr}(\mathcal{U})\rightarrow\text{Obj}(\mathcal{V})$ such that $\tau(U)=V$ create $A=\bigoplus_{X\in\text{Irr}(\mathcal{U})}X'\otimes\tau(X)$. Thus under certain conditions, $\tau$ extends only $A$ conformal extending $U\otimes V$. As examples, Kazhdan-Lusztig at generic levels obtain new product two affine algebras, prove equivalences $W$-algebras admissible levels.