Lemma 4.4.13. For a colimit-preserving functor \(F\colon \Sp \to C\), there exists a natural isomorphism \[ F(-) \, \cong \, - \otimes F(\S ). \] Moreover, this natural isomorphism can be chosen naturally in \(F \in \Fun ^{\colim }(\Sp ,C)\).
Proof. For objects \(Y,Z \in C\), there are natural isomorphisms \[ \hom _C(Y,Z) \overset {\text{Lemma 4.4.8}}{\cong } \hom _{\Sp }(\S , \hom _C(Y,Z)) \cong \hom _C(\S \otimes Y, Z), \] hence by the stable Yoneda lemma we obtain a natural isomorphism \(\S \otimes Y \cong Y\). □
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