It is somewhat of a standard construction, that, given a simplicial set K, we can form $E_n K$, by choosing a basepoint, and picking all simplices that map their $(n-1)$-skeleton to the basepoint. If K is a Kan complex, and is $(n-1)$-connected, then $E_n K \to K$ is a weak equivalence, and since $E_n K $ is also a Kan complex, it is a homotopy equivalance. My question is, given a simplicial map of $(n-1)$ connected simplicial sets $K\to L$, is it possible to choose homotopy inverses $K\to E_n K$, and $L\to E_n L$, such that the obvious diagram commutes?