Notation 4.6.9.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a (nonempty) finite sequence of vertices $X_0, X_1, \ldots , X_ n$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_{0}, X_1, \cdots , X_ n)$ denote the simplicial set given by the fiber product
\[ \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 0, 1, \cdots , n \} , \operatorname{\mathcal{C}}) } \{ (X_0, X_1, \cdots , X_ n) \} . \]