The Morita equivalence between parametrized spectra and module spectra

We give a Quillen equivalence between May and Sigurdsson’s model category of parametrized spectra over BG, and Mandell, May, Schwede, and Shipley’s model category of modules over the orthogonal ring spectrum Σ^∞ ₊G, for each topological group G. More generally, for a topological category C we introduce an “aggregate” model structure on the category of diagrams of spectra indexed by C, and prove that it is Quillen equivalent to spectra over BC. This strengthens earlier results that hold at the level of ∞-categories, and allows us to give a simple argument that the “dualizable” parametrized spectra are precisely the homotopy retracts of the finite cell spectra.