ScalingStacks

We have a string of equivalences: \[ \begin{aligned} U(\mu(X_1, \cdots, X_n)) & \simeq U(\mu( \mathrm{colim}_{[n_1] \in \Delta^{\mathrm{op}}} X'_1([n_1]), \cdots, \mathrm{colim}_{[n_m] \in \Delta^{\mathrm{op}}} X'_m([n_m]))) \\ & \simeq U(\mathrm{colim}_{([n_1], \cdots, [n_m]) \in {(\Delta^{\mathrm{op}})}^m} \mu(X'_1([n_1]), \cdots, X'_m([n_m])))\\ & \simeq U(\mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} \mu(X'_1([n]), \cdots, X'_m([n])))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} U(\mu(X'_1([n]), \cdots, X'_m([n])))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} U(\mu(FT^nU(X_1), \cdots, FT^nU(X_m)))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} UF(\mu(T^nU(X_1), \cdots, T^nU(X_m)))\\ & \simeq \mathrm{colim}_{[n]\in \Delta^{\mathrm{op}}} T(\mu(T^nU(X_1), \cdots, T^nU(X_m))) ~, \end{aligned}\] in which the third line uses the fact that the diagonal \(\Delta^{\mathrm{op}}\) in \({(\Delta^{\mathrm{op}})}^n\) is cofinal, which is equivalent to the statement that \(\Delta^{\mathrm{op}}\) is sifted [Lur17, Def. 5.5.8.1, Lem. 5.5.8.4]. For \(1 \leq i \leq n\), \(U(X_i) \in \mathcal L\) by asssumption. It follows that \(T^nU(X_i)\) is also in \(\mathcal L\). Since the \(\mathcal O\)-monoidal structure on \(\mathcal C\) is compatible with the factorization system and \(T\) preserves \(\mathcal L\), we see that \(T(\mu(T^nU(X_1), \cdots, T^nU(X_m)))\) is in \(\mathcal L\). The result now follows from ([00L1]) and the fact that \(\mathcal L\) is closed under colimits. ◻