Of course we should define the pushout of simplicial objects as the pushout in the category of simplicial objects. This may lead to the potentially confusing situation where the pushouts exist but are not as described below. To avoid this we define the pushout directly as follows.
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:
$(V \amalg _ U W)_ n = V_ n \amalg _{U_ n} W_ n$,
$d^ n_ i = (d^ n_ i, d^ n_ i)$, and
$s^ n_ i = (s^ n_ i, s^ n_ i)$.
In other words, $V\amalg _ U W$ is the pushout of the presheaves $V$ and $W$ over the presheaf $U$ on $\Delta $.
Lemma 14.8.2. If $U, V, W$ are simplicial objects in the category $\mathcal{C}$, and if $a : U \to V$, $b : U \to W$ are morphisms and if $V\amalg _ U W$ exists, then we have for any fourth simplicial object $T$ of $\mathcal{C}$.
\[ \mathop{\mathrm{Mor}}\nolimits (V\amalg _ U W, T) = \mathop{\mathrm{Mor}}\nolimits (V, T) \times _{\mathop{\mathrm{Mor}}\nolimits (U, T)} \mathop{\mathrm{Mor}}\nolimits (W, T) \]
Proof. Omitted. $\square$
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