Let $K$ and $L$ be abstract simplicial complexes and let $V(K)$, $V(L)$ denote their vertex sets. Then a simplicial map $K \to L$ is a map $f\colon V(K)\to V(L)$ such that $\{v_0,\dots, v_n\}\in K$ implies $\{f(v_0),\dots,f(v_n)\}\in L$. Now an inverse map $g\colon V(L)\to V(K)$ needs not to be simpicial but there are many sources for the following criterion: If $f$ induces a bijection on the set of $q$-simplices then its its inverse map is simplicial.
I wonder if less suffices. Let $\{w_0,\dots,w_n\}\in L$ be an $n$-simplex. Then if $f$ is just surjective onto the set of (all) simplices in $L$ then we have a $k$simplex $\{v_0,\dots, v_k\}\in K$ with $\{f(v_0),\dots,f(v_k)\}=\{w_0,\dots,w_n\}$ and hence $\{g(w_0),\dots,g(w_n)\}=\{v_0,\dots,v_k\}\in K$ and $g$ is simplicial.
I have not used injectivity of $f$ nor that it maps between simplices of the same dimension. What did I miss or are these two conditions just superfluous?