The Stacks project

Definition 14.8.1. Let $\mathcal{C}$ be a category. Let $U, V, W$ be simplicial objects of $\mathcal{C}$. Let $a : U \to V$, $b : U \to W$ be morphisms. Assume the pushouts $V_ n \amalg _{U_ n} W_ n$ exist in $\mathcal{C}$. The pushout of $V$ and $W$ over $U$ is the simplicial object $V\amalg _ U W$ defined as follows:

1. $(V \amalg _ U W)_ n = V_ n \amalg _{U_ n} W_ n$,

2. $d^ n_ i = (d^ n_ i, d^ n_ i)$, and

3. $s^ n_ i = (s^ n_ i, s^ n_ i)$.

In other words, $V\amalg _ U W$ is the pushout of the presheaves $V$ and $W$ over the presheaf $U$ on $\Delta $.