Proposition 4.5.8.2. Let $X$ and $Y$ be simplicial sets, and let $c: X \times \Delta ^1 \times Y \rightarrow X \star Y$ denote the collapse map of Construction 4.5.8.1. Then the commutative diagram of simplicial sets
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\begin{equation} \begin{gathered}\label{equation:diagram-of-collapse} \xymatrix@R =50pt@C=50pt{ (X \times \{ 0 \} \times Y) \coprod (X \times \{ 1\} \times Y) \ar [r] \ar [d]^{\pi _{X} \coprod \pi _{Y}} & X \times \Delta ^1 \times Y \ar [d]^{c} \\ ( X \times \{ 0\} ) \coprod ( \{ 1\} \times Y ) \ar [r]^-{( \iota _{X}, \iota _{Y} )} & X \star Y} \end{gathered} \end{equation}
is a categorical pushout square.