Last vertex map is carried by original simplices

Each nonempty face is finite, so it has a unique greatest vertex. If ${\sigma }_0⊊\cdots ⊊{\sigma }_q$ is a chain, every $\max {\sigma }_i$ lies in ${\sigma }_q$. Their set is therefore a face of $K$, proving simpliciality. If the entire chain lies over a face $\sigma$, all these vertices belong to $\sigma$, proving the carrier assertion. Repeated selected vertices are permitted for a simplicial map.

The induced chain map sends an oriented chain to its ordered list of selected vertices if distinct, and to zero otherwise. It commutes with the ordinary boundary by the induced-chain-map lemma. In degree zero each vertex is sent to a vertex, so both augmentations equal $1$; extending by the identity in degree $-1$ gives an augmented chain map. The empty complex gives the empty vertex map and identity only in degree $-1$. No existence of a total order on an arbitrary set is inferred: the order is supplied data.