Compactly generated categories as Ind-completions

Proposition 22.3.7 (Compactly generated categories as Ind-completions, [Lurie (2009), Propositions 5.5.7.8 and 5.5.7.10]). Let \(D\) be an \(\infty \)-category with small colimits and assume that \(D^\omega \) is small. The following are equivalent:

(1)

Every object of \(D\) is a filtered colimit of compact objects.

(2)

The restricted Yoneda functor induces an equivalence \[ D\iso \Ind (D^\omega ). \]

Under these conditions, a colimit-preserving functor out of \(D\) is uniquely determined by its restriction to \(D^\omega \).