Warning 4.3.4.14. Let $X$ and $Y$ be simplicial sets, and let $X \diamond Y$ denote the simplicial set given by the iterated coproduct

Since the formation of geometric realization commutes with the formation of colimits, we have an evident comparison map of topological spaces

This map is always bijective, and is a homeomorphism if either $X$ or $Y$ is finite (see Corollary 3.5.2.2). In this case, Corollary 4.3.4.13 supplies a homeomorphism of geometric realizations $| X \diamond Y | \simeq | X \star Y |$. Beware that this homeomorphism does *not* arise from a morphism of simplicial sets. In the case $X = \Delta ^ p$ and $Y = \Delta ^{q}$, it arises from the homotopy

appearing in Example 4.3.4.7, which is not piecewise-linear with respect to the natural triangulation of the polysimplex $| \Delta ^{p} | \times | \Delta ^1 | \times | \Delta ^ q |$.