Corollary 4.3.4.13. 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.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.3.4.13. 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 3.5.1.8). Each of these maps is bijective (even a homeomorphism), by virtue of Proposition 4.3.4.11.
$\square$