Corollary 16.5.11. There is a unique equivalence \[ \Sp \simeq \Sp ^{\exc }(\An )=\Exc _*(\An _*^{\fin },\An ) \] under which the two functors to \(\An \) given by \(\Omega ^\infty \) and evaluation at \(S^0\) agree.
Proof. By Proposition 4.3.16, Proposition 16.5.10, both sides, equipped with the displayed functors to \(\An \), are stabilizations of \(\An \). The result follows from the uniqueness of stabilization. □
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