Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_0^{\tau}$ be the $\tau$-homotopy sheaves for which $\pi_0^{\tau}(X)$ is the sheafification of the presheaf $U\mapsto [U,X] $, where $[-,-]$ is the homotopy class of the $\tau$-local model category (the hom-set in the homotopy category).

Does $\pi_0^{\tau}$ commute with group completion $\Omega B(-)$? $\pi_0^{\tau}$ commutes with $\Omega$ since sheafification functor is exact and the hom-functor commute with limits, but how about the classifying space functor $B$?

Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_n^{\tau}$ be the $\tau$-homotopy sheaves defined as follow

Does $\pi_n^{\tau}$ commute with group completion $\Omega B(-)$? $\pi_n^{\tau}$ commutes with $\Omega$ since sheafification functor is exact and the hom-functor commute with limits, but how about the classifying space functor $B$?

Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_n^{\tau}$ be the $\tau$-homotopy sheaves. Does $\pi_n^{\tau}$ commute with group completion $\Omega B(-)$?

Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_0^{\tau}$ be the $\tau$-homotopy sheaves for which $\pi_0^{\tau}(X)$ is the sheafification of the presheaf $U\mapsto [U,X] $, where $[-,-]$ is the homotopy class of the $\tau$-local model category (the hom-set in the homotopy category). 

Does $\pi_0^{\tau}$ commute with group completion $\Omega B(-)$? $\pi_0^{\tau}$ commutes with $\Omega$ since sheafification functor is exact and the hom-functor commute with limits, but how about the classifying space functor $B$?