ScalingStacks

Observe that ${\mathcal{F}}_{\kappa}$ is a composite functor

(9.3)

$$\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}_{\omega}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex(\!\kappa)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex}.$$

By construction and Propositions 5.6 and 5.9, we have an exact sequence

$${\mathcal{A}}\longrightarrow{\mathcal{F}}_{\kappa}{\mathcal{A}}\longrightarrow\Sigma_{\kappa}{\mathcal{A}},$$

which is natural in small stable $\infty$-categories ${\mathcal{A}}$. Next, we check that ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ satisfies property (i) above.