The Fibered Meta-Isomorphism Conjecture for spaces-to-spectra functors with coefficients

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ respect weak equivalences and disjoint unions. Let $G$ be a group and ${\mathcal F}$ a family of subgroups of $G$. For a $G$-CW-complex $Z$, let $H_*^?(-;{\mathbf S}^{\downarrow G}_Z)$ denote the associated equivariant homology theory over $G$. Fibered Meta-Isomorphism Conjecture for a functor from spaces to spectra with coefficients. For every $G$-CW-complex $Z$, the equivariant homology theory $H_*^?(-;{\mathbf S}^{\downarrow G}_Z)$ over $G$ satisfies the Fibered Meta-Isomorphism Conjecture for $(G,\operatorname{id}_G)$ over $G$ and the family ${\mathcal F}$. This is the fibered coefficient version of the assembly conjecture; the supplied text gives no resolution status.