  We construct a Spanier-Whitehead type duality functor relating finite
$\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove
that every $\mathcal{C}$-homology theory is given by taking the homotopy groups
of a balanced smash product with a fixed $\mathcal{C}^{\mathrm{op}}$-spectrum.
We use this to construct Chern characters for certain rational
$\mathcal{C}$-homology theories.
