Suppose I have a higher category, say a simplicial category $C$, and I want to invert a certain type of morphism $W$. Simplicial localization (for example hammock localization), gives a localized simplicial categetory $W^{-1}C$. Then I truncate this category by passing to $\pi_0$ of hom-sets to obtain the 1-category $\pi_0(W^{-1}C)$.

Conversely, I may pass to the truncated 1-category $\pi_0(C)$, and then localize, to get the 1-category $W^{-1}\pi_0(C)$.

I guess there is a functor $\pi_0(W^{-1}C)\to W^{-1}\pi_0(C)$. Is it an equivalence of categories? I am also interested in 2-category truncations defined by $\Pi_1$ of hom-sets.

If it is not true, are there easy to understand counterexamples?