ScalingStacks

Corollary 10.4. We have an equivalence of spectra

$$\Nat(K(-),THH(-))\cong THH({\mathcal{S}}^{\omega})\simeq THH(\mathbb{S})\simeq\mathbb{S}.$$

Passing to $\pi_{0}$ on both sides we obtain an isomorphism between homotopy classes of natural transformations and $\pi_{0}(\mathbb{S})\cong{\mathbb{Z}}$.