Suppose that C is a fusion category (over the complex numbers) and that Z(C) is its Drinfel'd center. By definition an object in Z(C) consists of an object V in C together with a collection of half-braidings $V \otimes W \rightarrow W \otimes V$ for every object W in C satisfying some naturality conditions. Hence there is a restriction functor R:Z(C)->C given by forgetting the half-braiding. Adjoint to this is an induction functor I:C->Z(C).

A Theorem of Etingof-Nikshych-Ostrik says that $R(I(V)) = \bigoplus_X X \otimes V \otimes X^{\*}$. In particular, we see that I(V) is $\bigoplus_X X \otimes V \otimes X^{\*}$ (where X ranges over simple objects up to isomorphism) together with some particular choice of half-braidings. I'm pretty sure I know what those half-braidings are. In particular there's a nice picture (the X,Y summand of the half-braiding with W is a sum over diagrams with a trivalent vertex connecting W to X* and Y* and another trivalent vertex connecting X and Y to W, where the two vertices range over dual bases).

What I really would like is a reference that explains this so that I don't have to write it up myself. The only description I know is in ENO's "On Fusion Categories" where it's written in terms of weak Hopf algebras.

The motivation is removing any mention of weak Hopf algebras from the construction in Section 5 (about cyclotomicity of certain Drinfel'd centers) in Scott and my Noncyclotomic Fusion Categories. It turns out that the diagram description above can be slightly modified (in a way suggested to me by Ben Webster) in order to give a description of I(V) where V is an object in a non-split fusion category over an arbitrary field.