$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 1.1.6.8. Let $X$ and $Y$ be simplicial sets. If $X$ has dimension $\leq 1$, then the canonical map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, Y ) \rightarrow \operatorname{Hom}_{ \operatorname{Graph}}( \mathrm{Gr}( X), \mathrm{Gr}( Y) ) \]
is bijective.
Proof.
Set $G = \mathrm{Gr}(X)$. If $X$ has dimension $\leq 1$, then Proposition 1.1.4.12 supplies a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \underset { e \in \operatorname{Edge}(G )}{\coprod } \operatorname{\partial \Delta }^{1} \ar [r] \ar [d] & \underset { e \in \operatorname{Edge}(G)}{\coprod } \Delta ^{1} \ar [d] \\ \underline{ \operatorname{Vert}(G) } \ar [r] & X. } \]
It follows that, for any simplicial set $Y = Y_{\bullet }$, we can identify $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, Y )$ with the fiber product
\[ ( \underset { e \in \operatorname{Edge}(G)}{\prod } Y_1) \underset { \prod _{ e \in \operatorname{Edge}(G)} (Y_0 \times Y_0) }{\times } (\underset { v \in \operatorname{Vert}(G) )}{\prod } Y_0), \]
which parametrizes morphisms of directed graphs from $\mathrm{Gr}( X )$ to $\mathrm{Gr}( Y )$.
$\square$