Corollary 7.5.8.5. Suppose we are given a commutative diagram of simplicial sets
7.55
\begin{equation} \begin{gathered}\label{equation:categorical-pushout-vs-categorical-colimit} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_{0} \ar [d] \\ A_{1} \ar [r] & A_{01}, } \end{gathered} \end{equation}
which we identify with a functor $\mathscr {F}: [1] \times [1] \rightarrow \operatorname{Set_{\Delta }}$. Then (7.55) is a categorical pushout square (in the sense of Definition 4.5.4.1) if and only if $\mathscr {F}$ is a categorical colimit diagram (in the sense of Definition 7.5.8.2).