  We describe new structure on the Goodwillie derivatives of a functor, and we
show how the full Taylor tower of the functor can be recovered from this
structure. This new structure takes the form of a coalgebra over a certain
comonad which we construct, and whose precise nature depends on the source and
target categories of the functor in question. The Taylor tower can be recovered
from standard cosimplicial cobar constructions on the coalgebra formed by the
derivatives. We get from this an equivalence between the homotopy category of
polynomial functors and that of bounded coalgebras over this comonad.
  For functors with values in the category of spectra, we give a rather
explicit description of the associated comonads and their coalgebras. In
particular, for functors from based spaces to spectra we interpret this new
structure as that of a divided power right module over the operad formed by the
derivatives of the identity on based spaces.
