Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 4.6.1.16. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{q} & \operatorname{\mathcal{C}}' \ar [d] \\ \operatorname{\mathcal{D}}\ar [r]^-{ \overline{F} } & \operatorname{\mathcal{D}}'. } \]

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 $\operatorname{\mathcal{D}}$. Then composition with $F$ induces an isomorphism of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( F(X), F(Y) )_{ \overline{F}(e) }. \]