ScalingStacks

Proof. This is a consequence of [52, 4.2.4.4]. Given a diagram of small stable $\infty$-categories, the equivalence in theorem 4.22 gives rise to a diagram in the localization of $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$. Including the localization into $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$, we now obtain a diagram in $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\simeq\mathrm{N}((\Cat_{\mathcal{S}})^{\cf})$ and we can use [52, 4.2.4.4] to lift this to a rigid diagram in $\Cat_{\mathcal{S}}$. ∎