  Working over a field $k$ of characteristic zero, the category of analytic
contravariant functors on the category of finitely-generated free groups is
shown to be equivalent to the category of representations of the $k$-linear
category associated to the Lie operad.
  Two proofs are given of this result. The first uses the original
Ginzburg-Kapranov approach to Koszul duality of binary quadratic operads and
the fact that the category of analytic contravariant functors is Koszul.
  The second proof proceeds by making the equivalence explicit using the
$k$-linear category associated to the operad encoding unital associative
algebras, which provides the `twisting bimodule'. A key ingredient is the
Poincar\'e-Birkhoff-Witt theorem.
  Using the explicit formulation, it is shown how this equivalence reflects the
tensor product on the category of analytic contravariant functors, relating
this to the convolution product for representations of the category associated
to the Lie operad.
