  Generalized Temperley-Lieb-Jones (TLJ) 2-categories associated to weighted
bidirected graphs were introduced in unpublished work of Morrison and Walker.
We introduce unitary modules for these generalized TLJ 2-categories as strong
*-pseudofunctors into the *-2-category of row-finite separable bigraded Hilbert
spaces. We classify these modules up to *-equivalence in terms of weighted
bi-directed fair and balanced graphs in the spirit of Yamagami's classification
of fiber functors on TLJ categories and DeCommer and Yamashita's classification
of unitary modules for Rep(SUq(2)).
