Remark 4.3.3.15. Let $X$ and $Y$ be simplicial sets. For each $n$-simplex $\sigma : \Delta ^ n \rightarrow X \star Y$, exactly one of the following conditions holds:

The morphism $\sigma $ factors through $X$ (where we identify $X$ with a simplicial subset of $X \star Y$ as in Remark 4.3.3.14).

The morphism $\sigma $ factors through $Y$ (where we identify $Y$ with a simplicial subset of $X \star Y$ as in Remark 4.3.3.14).

The morphism $\sigma $ factors as a composition

\[ \Delta ^{n} = \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y, \]for integers $p,q \geq 0$ satisfying $p+1+q=n$ and simplices $\sigma _{-}: \Delta ^{p} \rightarrow X$ and $\sigma _{+}: \Delta ^{q} \rightarrow Y$ of $X$ and $Y$, respectively. Moreover, in this case, the simplices $\sigma _{-}$ and $\sigma _{+}$ (and the integers $p,q \geq 0$) are uniquely determined.