The sphere spectrum acts as the unit

Proof. By the Yoneda lemma, it suffices to show that \(\Hom _{C}(- \otimes -,Z)\colon \Sp \catop \times C\catop \to \An \) preserves limits in both variables, for all \(Z \in C\). This functor is naturally isomorphic to \(\Hom _{\Sp }(-,\hom _{C}(-,Z))\). This clearly preserves limits in the first variable. For the second variable, this follows from the fact that \(\Omega ^{\infty }\hom _C(-,Z) = \Hom _C(-,Z)\colon C\catop \to \An \) preserves limits. □