Consider in the category sSet of simplicial sets, let $X$ be a $J$-indexed diagram in sSet ($J$ a small category), and for each $j\in J$, $X_j$ is a Kan complex, then do we have
$\pi_0({\rm holim}_{j\in J} X_j)\xrightarrow{\cong}{\rm lim}_{j\in J} \pi_0(X_j)$?
I can show this for $J$ being the poset $\mathbb{N}^{\rm op}$ as one can replace the diagram in such a way that all the maps in the (replaced) diagram are fibration, due to its simple shape.
Is it true in general?
(For a Kan complex $K$, $\pi_0K$ is the (reflexive) coequalizer of the two faces $K_1\to K_0$. Reflexive coequalizers in Set commute with finite products.)