TITLE: Stable ∞-categories as spectral categories
QUESTION [13 upvotes]: Let C be a stable ∞-category in the sense of Lurie's DAG I.  (In particular I do not assume that C has all colimits.)  Then C does have all finite colimits, the suspension functor on C is an equivalence, and C is enriched in Spectra in a way I don't want to make too precise (basically the Hom functor Cop × C → Spaces factors through Spectra and there are composition maps on the level of spectra).
Now suppose instead that C is an ∞-category which has all finite colimits and comes equipped with an enrichment in Spectra in the above sense.  One can show easily that C then has a zero object which allows us to define a suspension on C.  Suppose it is an equivalence.  Is C then a stable ∞-category?  Moreover, is the enrichment on C the one which comes from the fact that it is a stable ∞-category?

REPLY [8 votes]: According to Corollary 8.28 in DAG I a pointed $\infty$-category is stable iff it has finite colimits and the suspension functor is an equivalence.