ScalingStacks

The full subcategory \(\mathrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})}) \subseteq \mathrm{Ar}(\mathrm{Cat}_{(\infty, {k})})\) is closed under products and hence defines a Cartesian symmetric monoidal subcategory. Since all functors in  ([00BC]) preserve products, the pullback square is a pullback square of Cartesian symmetric monoidal \(\infty\)-categories and hence induces a pullback square of \(\infty\)-categories: Under the equivalence of \(\infty\)-categories \(\mathrm{Ar}(\mathrm{Alg}_{\mathcal O}(\mathrm{Cat}_{(\infty, {k})})) \simeq \mathrm{Alg}_{\mathcal O}(\mathrm{Ar}(\mathrm{Cat}_{(\infty, {k})}))\), the full subcategory \(\mathrm{Ar}^{(n-1)}(\mathrm{Alg}_{\mathcal O}(\mathrm{Cat}_{(\infty, {k})}))\) on the \(\mathcal O\)-monoidal functors \(F\) whose underlying functors \(F_X\) are \((n-1)\)-faithful becomes identified with \(\mathrm{Alg}_{\mathcal O}\left(\mathrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})})\right)\). Hence, the pullback square ([00BH]) is equivalent to the square and taking fibers at the \(\mathcal O\)-algebra \(\mathcal D\in\mathrm{Alg}_{\mathcal O} \left(\mathrm{Cat}_{(\infty, {k})}\right)\) induces the desired equivalence. ◻