  Morphisms in the linear category A of Jacobi diagrams in handlebodies give
rise to interesting contravariant functors on the category gr of
finitely-generated free groups, encoding part of the composition structure of
the category A. These functors correspond, via an equivalence of categories
given by Powell, to functors given by beaded open Jacobi diagrams. We study the
polynomiality of these functors and whether they are outer functors. These
results are inspired by and generalize previous results obtained by Katada.
