Let $f \colon X \to NC$ be a natural transformation. ($C$ a category and $N$ is nerve functor from Cat to sSet) Consider post-composition map $Hom(\wedge^n_1, X) \to Hom(\wedge^n_1, NC)$. Assume that the post composition map $Hom(\triangle^n, X) \to Hom(\triangle^n, NC)$ is bijective for $j < n$. Then show that $Hom(\wedge^n_1, X) \to Hom(\wedge^n_1, NC)$ is also bijective.
The problem comes from part of this proof: 
I get that the general idea is $\wedge^n_1$ is colimit of simplices of diemension less than $n$ and we can use this fact, but the details escape me. Any help would be appreciated, thanks!