Theorem 4.4.14 (Universality of \(\Sp \)). Let \(C\) be a stable \(\infty \)-category with small colimits. Then evaluation at the sphere spectrum induces an equivalence \[ \ev _{\S } \colon \Fun ^{\colim }(\Sp ,C) \quad \iso \quad C. \]
Proof. We first produce a natural isomorphism \(F(\S [-]) \cong \S [-] \otimes F(\S )\) of functors \(\An \to C\). In light of the equivalence \(\ev _*\colon \Fun ^{\colim }(\An ,C) \iso C\), it suffices to produce an isomorphism \(F(\S ) \cong \S \otimes F(\S )\), for which we take the one from Lemma 4.4.12.
Now, for computing \(F\) on an arbitrary spectrum \(X\), we use the standard presentation from Corollary 4.3.30. Using that \(F\) preserves colimits and shifts, we then get \[ F(X) \simeq F(\colim _n \Sigma ^{\infty -n} X_n) \simeq \colim _n \Omega ^n F( \Sigma ^{\infty } X_n) \simeq \colim _n \Omega ^n \cofib (F(\S ) \to F(\S [X_n])), \] where the last equivalence uses \(\Sigma ^{\infty }X_n \simeq \cofib (\S \to \S [X_n])\). Applying the same reasoning to \(F' = - \otimes F(\S )\), we similarly obtain \[ X \otimes F(\S ) \simeq \colim _n \Omega ^n \cofib (F(\S ) \to \S [X_n] \otimes F(\S )). \] The claim now follows from the identification \(F(\S [X_n]) \cong \S [X_n] \otimes F(\S )\) established before. These equivalences are natural in both \(F\) and \(X\): the standard presentation is natural in \(X\), while the initial identification of colimit-preserving functors \(\An \to C\) is natural in \(F\). We therefore obtain the claimed natural isomorphism in \(\Fun ^{\colim }(\Sp ,C)\). □
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