We will discuss the role of traces in the theory of Hopf-cyclic coefficients, and in particular use them to explain the appearance of the annuli stacking monoidal product on the coefficients of a braided category, such as appears, for example when the Hopf algebra is quasi-triangular. The traces also explain why the duals of the braided category, which need not be braided, also have this product on their coefficients; more precisely, what is the structure that the duals possess that remembers the structure that the braiding introduced?