## Representing map of x

Proposition:
(Representing map of $x$). Let $X$ be a simplicial set and let $x\in X_n$. There exists a unique simplicial map $f_x:\Delta[n]\to X$ such that $f_x(\sigma_n)=x$.
Proof:
Let $(i_0,i_1,\dots,i_k)\in\Delta[n]$. $(i_0,i_1,\dots,i_k)$ can be written as iterated compositions of faces and degeneracies of $\sigma_n$, i.e. $(i_0,i_1,\dots,i_k)=g\sigma_n$ where $g$ is an iterated composition of faces $(d_j)$ and degeneracies ($s_j$). Then
\begin{aligned} f_x(i_0,i_1,\dots,i_k)&=f_x(g\sigma_n)\\ &=gf_x(\sigma_n)\\ &=gx \end{aligned}
This defines a unique simplicial map $f_x$ such that $f_x(\sigma_n)=x$.