ScalingStacks

Theorem 6.7. The functor ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ inverts Morita equivalences, preserves filtered colimits, and sends split-exact sequences in $\Cat_{\infty}^{\ex}$ to cofiber sequences in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$. Moreover, ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ is universal with respect to these properties, i.e., given any pointed presentable $\infty$-category ${\mathcal{D}}$, we have an equivalence of $\infty$-categories

$$({\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}})^{\ast}:\mathrm{Fun}^{\mathrm{L}}({\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}},{\mathcal{D}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Fun}_{\mathrm{add}}^{\mathrm{un}}(\Cat_{\infty}^{\ex},{\mathcal{D}})\,,$$

where the right-hand denotes the full subcategory of $\mathrm{Fun}(\Cat_{\infty}^{\ex},{\mathcal{D}})$ of morphisms of $\infty$-categories which satisfy the above conditions.