ScalingStacks

Proof. By A.4.7.3.11, the adjunction $F\dashv U$ is monadic. Hence, given a morphism $A\to B$ in $\mathcal{C}$, by 2.4.6 the morphism $T\left(A\right)\to T\left(B\right)$ can be expressed as

$$\coprod_{n\geq 0}\left(P\left(n\right)\otimes A^{\otimes n}\right)_{h\Sigma_{n}}\to\coprod_{n\geq 0}\left(P\left(n\right)\otimes B^{\otimes n}\right)_{h\Sigma_{n}}.$$

By 4.4.2, $n$-connected morphisms are closed under $\otimes$ and, by T.5.2.8.6, they are closed under colimits. Hence, we obtain that $T\left(A\right)\to T\left(B\right)$ is $n$-connected as well. ∎