Level-rank duality via tensor categories

We give a new way to derive branching rules for the conformal embedding $$({\text{asl}}_n{)}_m\oplus ({\text{asl}}_m{)}_n\subset ({\text{asl}}_{nm}{)}_1\mathrm{.}$$ In addition, we show that the category $\text{Cc}({\text{asl}}_n{)}_m^0$ of degree zero integrable highest weight $({\text{asl}}_n{)}_m$-representations is braided equivalent to $\text{Cc}({\text{asl}}_m{)}_n^0$ with the reversed braiding.