Stabilization of an ∞-category

Definition 4.3.1. A stabilization of an \(\infty \)-category \(C\) with finite limits is a stable \(\infty \)-category \(\Sp (C)\) equipped with a left exact functor \(\Omega ^{\infty }\colon \Sp (C) \to C\) such that for every stable \(\infty \)-category \(D\) the composition functor \[ \Omega ^{\infty } \circ -\colon \Fun ^{\ex }(D,\Sp (C)) \overset {}{=} \Fun ^{\lex }(D,\Sp (C)) \to \Fun ^{\lex }(D,C) \] is an equivalence, where the first equality holds by Corollary 4.2.27.