If $K$ is a simplicial set, we can extend it to a bisimplicial set $K^\prime: \Delta^{op} \to sSet$ by setting $K^\prime_n = (K_n)_\delta$ where $(-)_\delta$ means taking a discrete simplicial set. There is a weak equivalence $hocolim K^\prime \to K$.
If K is a finite-dimensional simplicial set, it is isomorphic to some $n$-skeleton, say $sk_n K \cong K$. The same is true for $K^\prime$. Denote by $\Delta^{op}_{\leq n}$ the full subcategory of $\Delta^{op}$ on the objects $[0],\ldots,[n]$ and by $i: \Delta^{op}_{\leq n} \to \Delta^{op}$ the inclusion. My question:
Is the map $hocolim \left( i^\ast (K^\prime) : \Delta^{op}_{\leq n} \to sSets \right) \to K$ a weak equivalence?
Edit: added some $op$'s