Does the nerve functor $Cat\to sSet$ from small categories to simplicial sets preserves weak equivalences?

If $f:C\to D$ and $g:D\to C$ are functors of small categories such that $\alpha:f\circ g\sim 1_D$ and $\beta:g\circ f \sim 1_C$. 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?)