Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 4.6.1.15. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$ having the same image $D = q(X) = q(Y)$ in $\operatorname{\mathcal{D}}$. Then we have a canonical isomorphism of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{ \operatorname{id}_{D} } \simeq \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{D} }(X,Y), \]

where $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ denotes the fiber of $q$ over the vertex $D$.