Show that give any map $f_0$ from the set of vertices of $\sigma$ to the set of vertices of $\tau$, where $\sigma$ and $\tau$ are simplices, there is a unique simplicial map $f:\sigma\to\tau$ whose restriction to the vertices of $\sigma$ is $f_0$.
This is a question from Lee's "Introduction to Topological Manifolds". Let $\sigma$, $\tau$ be two triangles. Going counter-clockwise from the left-most vertex, we have the vertices $123$ for $\sigma$ and $132$ for $\tau$. Can there really be an affine map (a simplicial map has to be an affine map) which maps $\sigma$ to $\tau$?
Hint: work inductively. A map on $0$-simplices induces a map on the $1$-simplices by sending $[a,b]$ to $[f_0(a),f_0(b)]$. Show that this extends to higher simplices and satisfies the properties needed to be a simplicial map.