Stable presentable categories as module categories

There is a theorem of Schwede and Shipley which classifies categories of modules over an A ring spectrum as those stable presentable (∞,1)-categories with a compact generator. Suppose I allow my A rings to "have many objects", that is, I consider categories of the form FunSp(Iop, Sp) where Sp is the category of spectra, I is a small Sp-enriched category (in some appropriate sense) and Funsp denotes the category of Sp-enriched functors. Is there a classification of which stable presentable categories can be obtained in this way? Is it possible that all stable presentable categories are of this form?

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