Limits of spectrum objects

Proof. By the pointwise criterion for adjunctions from Lemma 21.1.4, it suffices to show that for every stable \(\infty \)-category \(D\), composition with \(\Omega ^{\infty }\colon \Sp (C) \to C\) induces an equivalence \[ \Omega ^{\infty } \circ -\colon \Hom _{\Cat ^{\st }_{\infty }}(D,\Sp (C)) \iso \Hom _{\Cat ^{\lex }_{\infty }}(D,C). \] But since a functor \(D \to \Sp (C)\) is exact if and only if it is left exact, this map is obtained from the defining equivalence \(\Fun ^{\lex }(D,\Sp (C)) \iso \Fun ^{\lex }(D,C)\) by passing to groupoid cores. □