  We give two proofs of a level-rank duality for braided fusion categories
obtained from quantum groups of type $C$ at roots of unity. The first proof
uses conformal embeddings, while the second uses a classification of braided
fusion categories associated with quantum groups of type $C$ at roots of unity.
In addition we give a similar result for non-unitary braided fusion categories
quantum groups of types $B$ and $C$ at odd roots of unity.
