ScalingStacks

Proposition 8.5. The functor $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$ preserves $\kappa$-filtered colimits and sends strict-exact sequences to cofiber sequences in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$

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

$\displaystyle\mapsto$

$\displaystyle\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}})\longrightarrow\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{B}})\longrightarrow\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{B}}/{\mathcal{A}})\,.$

Moreover, $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$ is universal with respect to these two properties, i.e., given any stable presentable $\infty$-category ${\mathcal{D}}$, we have an equivalence of $\infty$-categories

$$(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}})^{\ast}\colon\mathrm{Fun}^{\mathrm{L}}(\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}},{\mathcal{D}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Fun}_{\underline{\mathrm{wloc}}}^{\kappa}(\Cat_{\infty}^{\perf},{\mathcal{D}})\,,$$

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