Proposition 16.5.10. For every \(\infty \)-category \(C\) with finite limits, the evaluation functor \(\ev _{S^0}\colon \Sp ^{\exc }(C) \to C\) exhibits \(\Sp ^{\exc }(C)\) as a stabilization of \(C\).
Proof. The functor \(\ev _{S^0}\) is left exact, since limits in the functor category are computed pointwise and reduced excisive functors are closed under limits. Let \(D\) be stable. Since pushout squares and pullback squares agree in \(D\), a functor \(D \to C\) is left exact if and only if it is reduced and excisive. Thus it is enough to show that \[ \Exc _*(D,\Sp ^{\exc }(C)) \to \Exc _*(D,C) \] is an equivalence. Since reducedness and excisiveness are pointwise conditions, currying restricts to the relevant full subcategories and identifies the source with \[ \Exc _*(D,\Exc _*(\An _*^{\fin },C)) \simeq \Sp ^{\exc }(\Exc _*(D,C)). \] The target \(\Exc _*(D,C)\) is stable by Proposition 16.5.7, so evaluation at \(S^0\) is an equivalence by Lemma 16.5.8. □
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