ScalingStacks

Proof. By the Yoneda lemma it is enough to show that for every $n$-truncated object $Z$ in $\mathcal{C}$ the induced map

$$f_{*}\colon\operatorname{Map}\left(B,Z\right)\to\operatorname{Map}\left(A,Z\right)$$

is an equivalence. For this, it is enough to show that for every $g\colon A\to Z$, the fiber of $f_{*}$ over $g$ is contractible. By T.5.5.5.12, the fiber is equivalent to the space of lifts for the square

which is contractible by definition as $f\colon A\to B$ was assumed to be $n$-connected. ∎