Remark 1.1.6.10. The proof of Proposition 1.1.6.9 gives an explicit description of the inverse equivalence $\operatorname{Graph}\simeq \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Set_{\Delta }}$: it carries a directed graph $G$ to the $1$-dimensional simplicial set $G_{\bullet }$ given by the colimit of the diagram
\[ ( \underset { v \in \operatorname{Vert}( G )}{\coprod } \Delta ^0 ) \leftarrow (\underset { e \in \operatorname{Edge}( G )}{\coprod } \operatorname{\partial \Delta }^{1} ) \rightarrow ( \underset { e \in \operatorname{Edge}( G)}{\coprod } \Delta ^{1} ). \]