Definition 4.6.4.1 (The Oriented Fiber Product). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. We let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the simplicial set given by the iterated fiber product
\[ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}(\{ 1\} , \operatorname{\mathcal{E}}) } \operatorname{\mathcal{D}}. \]
We will refer to $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ as the oriented fiber product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ over $\operatorname{\mathcal{E}}$.