ScalingStacks

Proof. The result follows from theorem 6.7 and from the universal property of stabilization (i.e., [53, 1.4.5.5]). Note that stabilization preserves colimits and ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ sends split-exact sequences to cofiber sequences, so the split-exact sequence (6.2) is sent to a split cofiber sequence ${\mathcal{U}}_{\mathrm{add}}({\mathcal{C}})\simeq{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}})\vee{\mathcal{U}}_{\mathrm{add}}({\mathcal{B}})$ in ${\mathcal{M}}_{\mathrm{add}}$. ∎