Does the nerve functor $\operatorname{Cat}\to \operatorname{sSet}$ from small categories to simplicial sets preserves 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 $\beta\colon g\circ f \sim 1_C$. We have a pair of maps between the corresponding simplicial sets. What can we say on $\alpha$ and $\beta$ under the nerve functor? (Homotopic in some sense?)