I am reading Lurie's Elliptic Cohomology II and it claims (Section 4.1.3) that for an $\mathbb{E}\_\infty$-ring $A$ "there is an essentially unique symmetric monoidal functor $\mathcal{S} \to \operatorname{Mod}\_A$ which preserves small colimits", where $\mathcal{S}$ is the category of spaces.


At first I thought this would be the map $X \mapsto \Sigma^\infty X \wedge A$. As spectra, it preserves small colimits, but I am not sure this is the case as $A$-modules. Does anyone have a more explicit description of this functor?



Monoidal colimit-preserving functor from spaces to $A$-modules
Your description of the functor is correct. The suspension spectrum functor is homotopy colimit preserving, as you say. Smashing with A is left adjoint to the forgetful functor from A-modules to spectra, and this means that it also preserves homotopy colimits.


