ScalingStacks

Proof. Any stably representable functor $\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is exact, giving the factorization

$$\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})\subseteq\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty}).$$

By proposition 3.2, we may rewrite this as $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind(\mathcal{C})\simeq\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$ to see that, as an exact functor $\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$, any stably representable functor is also compact. ∎