ScalingStacks

Proof. The fact that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ preserves $\kappa$-filtered colimits is clear. Since a functor ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a Morita equivalence if and only if $\Idem({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Idem({\mathcal{B}})$ is an equivalence, proposition 5.31 allow us to conclude that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ inverts Morita equivalences. We now show that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ sends exact sequences to cofiber sequences. Let

$${\mathcal{A}}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{C}}$$

be an exact sequence. Since we have an induced Morita equivalence ${\mathcal{B}}/{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$, it suffices to show that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ sends exact sequences of shape

$${\mathcal{A}}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{B}}/{\mathcal{A}}$$

to cofiber sequences in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$. Since $\Idem:\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}$ is a localization, it commutes with colimits and therefore the right-hand vertical map in the diagram

is a Morita equivalence. The bottom line is a strict-exact sequence, and so we conclude that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ sends exact sequences to cofiber sequences. Finally, the universality of ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ follows from propositions 8.3 and 8.5, and from the universal property of localization (see section 2.5). ∎