ScalingStacks

Lemma 4.2.5. Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories that admit finite limits and let $F\colon\mathcal{C}\leftrightarrows\mathcal{D}\penalty\mskip 6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0muG$ be an adjunction with $F\dashv G$,

1. (1)

   For every $d\geq-2$ and a $d$-truncated morphism $g$ in $\mathcal{D}$, the morphism $G\left(g\right)$ is a $d$-truncated morphism in $\mathcal{C}$.

2. (2)

   For every $n\geq-2$ and an $n$-connected morphism $f$ in $\mathcal{C}$, the morphism $F\left(f\right)$ is an $n$-connected morphism in $\mathcal{D}$.