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Feb 22, 2018 at 12:04 comment added Denis Nardin No, the bijection is with left-exact localizations of $\mathrm{Psh}(\mathcal{C})$, which are special reflective subcategories (those for which the localization functor commutes with finite limits).
Feb 21, 2018 at 22:11 history answered Qfwfq CC BY-SA 3.0