Construction 3.4.2.2 (Homotopy Pushouts). Let $f_0: A \rightarrow A_0$ and $f_1: A \rightarrow A_1$ be morphisms of simplicial sets. We let $A_0 {\coprod }_{A}^{\mathrm{h}} A_{1}$ denote the iterated pushout
\[ A_0 \coprod _{(\{ 0\} \times A)} (\Delta ^1 \times A) \coprod _{ (\{ 1\} \times A) } A_{1}. \]
We will refer to $A_{0} {\coprod }_{A}^{\mathrm{h}} A_{1}$ as the homotopy pushout of $A_0$ with $A_1$ along $A$. Note that the projection map $\Delta ^1 \times A \twoheadrightarrow A$ induces a comparison map $A_0 {\coprod }_{A}^{\mathrm{h}} A_{1} \twoheadrightarrow A_0 {\coprod }_{A} A_1$ from the homotopy pushout to the usual pushout, which is an epimorphism of simplicial sets.