Construction 4.6.1.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a pair of vertices $X$ and $Y$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ denote the simplicial set given by the fiber product
\[ \{ X\} \times _{ \operatorname{Fun}(\{ 0\} , \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) } \{ Y\} . \]
We will typically be interested in this construction only in the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category; if this condition is satisfied, we will refer to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ as the space of morphisms from $X$ to $Y$.