ScalingStacks

Proof. It suffices to show that any spectrum object $A$ of $\mathcal{C}$ is of the form $\Sigma^{\infty}z$ for a uniquely determined object $z$ of $\mathcal{C}$. This follows from the fact that since ${\mathcal{C}}$ is stable, $\Omega^{\infty}\colon\mathrm{Sp}(\mathcal{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$ is an equivalence with inverse $\Sigma^{\infty}\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Sp}(\mathcal{C})$. ∎