Construction 4.3.4.9. Let $X$ and $Y$ be simplicial sets, and let $\sigma : \Delta ^ n \rightarrow X \star Y$ be a morphism. We define a continuous function $f(\sigma ): | \Delta ^ n | \rightarrow |X| \star |Y|$ as follows (see Remark 4.3.3.17):
If $\sigma $ factors through $X$, we let $f(\sigma )$ denote the composition
\[ | \Delta ^ n | \xrightarrow { | \sigma | } |X| \xrightarrow {\iota _{|X|}} |X| \star |Y|, \]where the second map is the inclusion of Remark 4.3.4.2.
If $\sigma $ factors through $Y$, we let $f(\sigma )$ denote the composition
\[ | \Delta ^ n | \xrightarrow { | \sigma | } |Y| \xrightarrow { \iota _{|Y|}} |X| \star |Y|, \]where the second map is the inclusion of Remark 4.3.4.2.
If $\sigma $ factors as a composition
\[ \Delta ^ n = \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y, \]then we let $f(\sigma )$ denote the composite map
\[ | \Delta ^ n | = | \Delta ^{p+1+q} | \xrightarrow { H_{p,q}^{-1} } | \Delta ^{p} | \star | \Delta ^{q} | \xrightarrow { | \sigma _{-} | \star | \sigma _{+} | } |X| \star |Y|, \]where $H_{p,q}$ denotes the homeomorphism of Proposition 4.3.4.8.
The construction $\sigma \mapsto f(\sigma )$ is compatible with face and degeneracy operators, and therefore determines a morphism of simplicial sets $f: X \star Y \rightarrow \operatorname{Sing}_{\bullet }( |X| \star |Y| )$. We will identify $f$ with a continuous function $T_{X,Y}: | X \star Y | \rightarrow |X| \star |Y|$, which we will refer to as the join comparison map.