Title: A characterization of cellular motivic spectra,
Abstract: Let $ \alpha: \mathcal{C} \to \mathcal{D}$ be a symmetric monoidal functor
from a stable presentable symmetric monoidal $\infty$-category $\mathcal{C} $
compactly generated by the tensorunit to a stable presentable symmetric
monoidal $\infty$-category $ \mathcal{D} $ with compact tensorunit. Let $\beta:
\mathcal{D} \to \mathcal{C}$ be a right adjoint of $\alpha$ and $ \mathrm{X}:
\mathcal{B} \to \mathcal{D} $ a symmetric monoidal functor starting at a small
rigid symmetric monoidal $\infty$-category $ \mathcal{B}$. We construct a
symmetric monoidal equivalence between modules in the $\infty$-category of
functors $ \mathcal{B} \to \mathcal{C} $ over the $ \mathrm{E}_\infty$-algebra
$\beta \circ \mathrm{X} $ and the full subcategory of $\mathcal{D}$ compactly
generated by the essential image of $\mathrm{X}$. Especially for every motivic
$ \mathrm{E}_\infty$-ring spectrum $\mathrm{A}$ we obtain a symmetric monoidal
equivalence between the $\infty$-category of cellular motivic
$\mathrm{A}$-module spectra and modules in the $\infty$-category of functors
$\mathrm{QS}$ to spectra over some $ \mathrm{E}_\infty$-algebra, where
$\mathrm{QS}$ denotes the 0th space of the sphere spectrum.