Let me first ask an intuitive version of the question:

Let $Sp$ be the homotopy category of spectra. Let $E$ be a ring spectrum. Let $$D:Sp \to Sp$$ be the Spanier-Whitehead dual functor (maybe we need to restrict ourselves to compact object and that's fine). I want to define an algebraic Spanier-Whitehead duality functor $$ D_E: E_*E\text{-}coMod \to E_*E\text{-}coMod $$ such that it commutes with the $E$-hurewicz map $h_E$, i.e. $$ h_E \circ D = D_E \circ h_E $$.

The question under what conditions on $E$ there exists $D_E$? For example, when $E = H\mathbb{F}_p$ we know that $D_E = Hom_{\mathbb{F}_p}(M, \mathbb{F}_p)$.

Now let me make this question a little more precise. Maybe we need to replace $E_*E\text{-}coMod$ with finite objects in its derived category.

My gut feeling says, that if $E$ satisfies

- Flatness: $E_*E$ is flat over $E_*$,
- Adams Condition: Check out Definition 3.1, pg 16 of https://sites.math.northwestern.edu/~pgoerss/papers/sum.pdf

then it might just be enough to construct $D_E$. Is it though? If so can someone sketch a proof? If not do I need additional conditions?

So now the question is how to define dual objects in the (derived) category of $E_*E$ comodule and under what conditions on $E$ is it compatible with the Hurewicz map.