Proposition 1.1.6.9. Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets and let $\operatorname{Set}_{\Delta }^{\leq 1} \subseteq \operatorname{Set_{\Delta }}$ denote the full subcategory spanned by the simplicial sets of dimension $\leq 1$. Then the construction $X \mapsto \mathrm{Gr}( X )$ induces an equivalence of categories $\operatorname{Set}_{\Delta }^{\leq 1} \rightarrow \operatorname{Graph}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Proposition 1.1.6.8 that the functor $X \mapsto \mathrm{Gr}(X )$ is fully faithful when restricted to simplicial sets of dimension $\leq 1$. It will therefore suffice to show that it is essentially surjective. Let $G$ be any directed graph, and form a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \underset { e \in \operatorname{Edge}( G )}{\coprod } \operatorname{\partial \Delta }^{1} \ar [r] \ar [d]^{(s,t)} & \underset { e \in \operatorname{Edge}( G)}{\coprod } \Delta ^{1} \ar [d] \\ \underset { v \in \operatorname{Vert}( G )}{\coprod } \Delta ^{0} \ar [r] & X. } \]
Then $X$ is a simplicial set of dimension $\leq 1$ (Proposition 1.1.3.11), and the directed graph $\mathrm{Gr}( X)$ is isomorphic to $G$. $\square$