 Solution

A simplicial approximation $g:K\to L$ to $f:|K|\to |L|$ is a vertex map satisfying the open-star conditionfor every vertex $v$. Here an open star consists of points whose barycentric coordinate at that vertex is positive. This condition ensures that the images of the vertices of every simplex span a simplex of $L$, and the vertex map extends linearly over simplices.

To triangulate $|K|\times |L|$, order the vertices of each complex consistently. For each product $[v_0,\dots ,v_p]\times [w_0,\dots ,w_q]$, use the simplices whose vertices $(v_i,w_j)$ follow monotone lattice paths from $(0,0)$ to $(p,q)$, taking a step in one coordinate at a time. These prism triangulations agree on faces and cover the product.

If $g,h$ both approximate $f$, choose $x$ in the relative interior of a simplex $\sigma$. Then $x$ belongs to the open star of every vertex $v$ of $\sigma$. Hence $f(x)$ belongs to every open star of $g(v)$ and $h(v)$. Its unique carrier simplex in $L$ contains all these vertices. Thus $g(\sigma )$ and $h(\sigma )$ are faces of that one simplex, proving that $g,h$ are contiguous simplicial maps.