Warning 1.1.6.6. Note that part $(b)$ of Definition 1.1.6.5 allows the possibility that a morphism of directed graphs $G \rightarrow G'$ can “collapse” edges of $G$ to vertices of $G'$. Many other notions of morphism between (directed) graphs appear in the literature; we single out Definition 1.1.6.5 because of its close connection with the theory of simplicial sets (see Proposition 1.1.6.7 below).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$