Kerodon

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Remark 4.6.1.18. 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 $\operatorname{\mathcal{D}}$. Form a pullback diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^1 \ar [r]^-{e} & \operatorname{\mathcal{D}}, }$

so that $X$ lifts uniquely to a vertex $\widetilde{X} \in \operatorname{\mathcal{C}}'$ lying over the vertex $0 \in \Delta ^1$, and $Y$ lifts uniquely to a vertex $\widetilde{Y} \in \operatorname{\mathcal{C}}'$ lying over the vertex $1 \in \Delta ^1$. Remark 4.6.1.17 and Example 4.6.1.15 supply isomorphisms

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( \widetilde{X}, \widetilde{Y} )_{ \operatorname{id}_{\Delta ^1} } = \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( \widetilde{X}, \widetilde{Y} ).$