Corollary Let $X$ and $Y$ be simplicial sets. Then the join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ is a bijection.

Proof. As a map of sets, we can realize $T_{X,Y}$ as a filtered colimit of join comparison maps $T_{X', Y}$, where $X'$ ranges over the finite simplicial subsets of $X$ (Remark Each of these maps is bijective (even a homeomorphism), by virtue of Proposition $\square$