ScalingStacks

Proof. First note that if each of the $b_{i}$ were a representable presheaf, then the map $A\to B$ may be written as a pushout of the generating morphism $\mathcal{T}_{n,\infty}$, hence is manifestly an element of $T_{\Theta_{n}}$ (we leave this as an exercise). The general case, however, reduces to this case as every presheaf is (canonically) a colimit of representables, the functors $\sigma^{[\ell]}_{!}$ commute with these colimits separately in each variable, and $T_{\Theta_{n}}$, being a saturated class, is closed under colimits. ∎