$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Let $X$, $Y$, and $K$ be simplicial sets. Unwinding the definitions, we see that morphisms from $K$ to $X \star Y$ can be identified with triples $(\pi , f_{-}, f_{+} )$, where

\[ \pi : K \rightarrow \Delta ^1 \quad \quad f_{-}: \{ 0\} \times _{\Delta ^1} K \rightarrow X \quad \quad f_{+}: \{ 1\} \times _{\Delta ^1} K \rightarrow Y \]

are morphisms of simplicial sets (note that, when $K$ is a simplex, this recovers the description of Remark