  Let $\mathcal{C}$ be a finite braided multitensor category. Let $B$ be
Majid's automorphism braided group of $\mathcal{C}$, then $B$ is a
cocommutative Hopf algebra in $\mathcal{C}$. We show that the center of
$\mathcal{C}$ is isomorphic to the category of left $B$-comodules in
$\mathcal{C}$, and the decomposition of $B$ into a direct sum of indecomposable
$\mathcal{C}$-subcoalgebras leads to a decomposition of
$B$-$\operatorname*{Comod}_{\mathcal{C}}$ into a direct sum of indecomposable
$\mathcal{C}$-module subcategories.
  As an application, we present an explicit characterization of the structure
of irreducible Yetter-Drinfeld modules over semisimple quasi-triangular weak
Hopf algebras. Our results generalize those results on finite groups and on
quasi-triangular Hopf algebras.
