Kerodon

Construction 4.6.1.15. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y)$ be an edge of the simplicial set $\operatorname{\mathcal{D}}$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e}$ denote the fiber product $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( q(X), q(Y) ) } \{ e\} $, which we regard as a simplicial subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.