Example 1.2.6.4. Let $G$ be a directed graph having a single vertex $\operatorname{Vert}(G) = \{ x\} $ and a single edge $\operatorname{Edge}(G) = \{ e \} $ (necessarily satisfying $s(e) = x = t(e)$). Then the path category $\operatorname{Path}[G]$ has a single object $x$ whose endomorphism monoid $\operatorname{End}_{\operatorname{Path}[G]}(x) = \operatorname{Hom}_{\operatorname{Path}[G]}(x,x)$ can be identified with the set $\operatorname{\mathbf{Z}}_{\geq 0}$ of nonnegative integers (with monoid structure given by addition).

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$