ScalingStacks

We now turn to condition ([00KF]) of Observation B.2.1: given any morphism \(\mathcal O\xrightarrow{h} \mathcal O'\) in \(\mathrm{Op}\), we must show that the factorization in \((\mathrm{Cat}_\infty)_{/\mathrm{Fin}_*}\) determined by its (\(n\)-surjective, \(n\)-faithful) factorization system in fact lies in \(\mathrm{Op}\). To simplify our notation, we write \(\mathcal F^{\otimes} \coloneqq \mathrm{Fact}(h)\). Additionally, we write \(\mathcal O^{\otimes} \xrightarrow{p} \mathrm{Fin}_*\), \(\mathcal O'^{\otimes} \xrightarrow{p'} \mathrm{Fin}_*\), and \(\mathcal F^{\otimes} \xrightarrow{q} \mathrm{Fin}_*\) for the indicated functors. We note immediately that the claim is trivial both when \(n = -2\) (since then \(\mathcal F^{\otimes} \xrightarrow{r} \mathcal O'^{\otimes}\) is an equivalence) and when \(n = -1\) (since then \(\mathcal F^{\otimes} \xrightarrow{r} \mathcal O'^{\otimes}\) is the inclusion of the full suboperad on the colors in the image of \(\underline{\mathcal O} \xrightarrow{\underline{h}} \underline{\mathcal O'}\)). So, we henceforth assume that \(n \geq 0\).