  Every fusion category C that is k-linear over a suitable field k, is the
category of finite-dimensional comodules of a Weak Hopf Algebra H. This Weak
Hopf Algebra is finite-dimensional, cosemisimple and has commutative bases. It
arises as the universal coend with respect to the long canonical functor
\omega:C->Vect_k. We show that H is a quotient H=H[G]/I of a Weak Bialgebra
H[G] which has a combinatorial description in terms of a finite directed graph
G that depends on the choice of a generator M of C and on the fusion
coefficients of C. The algebra underlying H[G] is the path algebra of the
quiver GxG, and so the composability of paths in G parameterizes the truncation
of the tensor product of C. The ideal I is generated by two types of relations.
The first type enforces that the tensor powers of the generator M have the
appropriate endomorphism algebras, thus providing a Schur-Weyl dual description
of C. If C is braided, this includes relations of the form `RTT=TTR' where R
contains the coefficients of the braiding on \omega M\otimes\omega M, a
generalization of the construction of Faddeev-Reshetikhin-Takhtajan to Weak
Bialgebras. The second type of relations removes a suitable set of group-like
elements in order to make the category of finite-dimensional comodules
equivalent to C over all tensor powers of the generator M. As examples, we
treat the modular categories associated with U_q(sl_2).
