Proposition 4.4.16. There exists a unique tensor product functor \[ - \otimes -\colon \Sp \times \Sp \to \Sp \] which preserves colimits in both variables and satisfies \(\S \otimes \S \cong \S \).
Proof. By Theorem 4.4.14, \(F\) is of the form \(- \otimes X\) for some \(X \in C\), hence has right adjoint given by the mapping spectrum functor \(\hom _C(X,-)\colon C \to \Sp \). □
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