ScalingStacks

Proof. Given a morphism $f\colon A\to B$ in $\mathcal{D}$, using the canonical simplicial resolution provided by the proof of A.4.7.3.13, we can express it as a colimit of the simplicial diagram of morphisms:

$$\operatorname*{colim}\limits_{\Delta^{op}}\left(T^{n+1}\left(A\right)\to T^{n+1}\left(B\right)\right),$$

which one can write as

$$\operatorname*{colim}\limits_{\Delta^{op}}\left(FT^{n}U\left(A\right)\to FT^{n}U\left(B\right)\right).$$

If $U\left(f\right)$ is $n$-connected as in the statement, then since $T$ preserves $n$-connected morphisms by assumption and $F$ preserves $n$-connected morphisms by being left adjoint, it follows that all the maps in the diagram are $n$-connected. By T.5.2.8.6(7), the map $f$ is also $n$-connected. ∎