The Drinfeld center $Z(C)$ of a braided category "contains" $C$ and $\bar C$ (the category with opposite braiding) and therefore also $C\boxtimes \bar C$ and you can show that the following is equivalent

1. $C$ is modular

2. $Z(C)$ is equivalent with $C\boxtimes \bar C$.

The Drinfeld center $Z(C)$ of a braided category "contains" $C$ and $\bar C$ (the category with opposite braiding) and therefore also $C\boxtimes \bar C$ and you can show that the following is equivalent

1. $C$ is modular

2. $Z(C)$ is equivalent with $C\boxtimes \bar C$.

In other words, for a braided category $C$ there is a natural notion of a center $Z(C)$ and a natural embedding $C\boxtimes \bar C$ in $Z(C)$ which is an equivalence if and only if $C$ is modular.