Proposition 4.4.9 (Tensoring by spectra). Let \(C\) be a stable \(\infty \)-category with small colimits. Then \(C\) is tensored over \(\Sp \): there exists a unique functor \[ - \otimes - \colon \Sp \times C \to C \] equipped with a natural equivalence of spectra \[ \hom _C(X \otimes Y, Z) \quad \cong \quad \hom _{\Sp }(X, \hom _C(Y,Z)) \] for \(X \in \Sp \) and \(Y,Z \in C\).
Proof. It suffices to show that these two functors agree after applying \(\Omega ^{\infty }\), and indeed we have \[ \Omega ^{\infty }\hom (\S ,-) \cong \Hom _{\Sp }(\S ,-) \cong \Hom _{\Sp }(\S [*],-) \cong \Hom _{\An }(*,\Omega ^{\infty }(-)) \cong \Omega ^{\infty }(-). \] Here we use that \(\Hom _{\An }(*,-)\colon \An \to \An \) is equivalent to the identity, since its left adjoint \(- \times *\colon \An \to \An \) is equivalent to the identity. □
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