Does the nerve functor $\operatorname{Cat}\to \operatorname{sSet}$ from small categories to simplicial sets preserve weak equivalences? 

If $f\colon C\to D$ and $g\colon D\to C$ are functors of small categories such that $\alpha\colon f\circ g\sim 1_D$ and $\beta\colon g\circ f \sim 1_C$, then we have a pair of maps between the corresponding simplicial sets. What can we say about $\alpha$ and $\beta$ under the nerve functor? (Homotopic in some sense?)