  Let M be a smooth manifold, and let O(M) be the poset of open subsets of M.
Let C be a category that has a zero object and all small limits. A homogeneous
functor (in the sense of manifold calculus) of degree k from O(M) to C is
called very good if it sends isotopy equivalences to isomorphisms. In this
paper we show that the category VGHF of such functors is equivalent to the
category of contravariant functors from the fundamental groupoid of Conf(k, M)
to C, where Conf(k, M) stands for the unordered configuration space of k points
in M. As a consequence of this result, we show that the category VGHF is
equivalent to the category of representations of the fundamental group of
Conf(k, M) in C, provided that Conf(k, M) is connected. We also introduce a
subcategory of vector bundles that we call very good vector bundles, and we
show that it is abelian, and equivalent to a certain category of very good
functors.
