Example 5

Let $\varphi : G\rightarrow G'$ be a morphism of groupoids. Let H be the subquiver of G with set of vertices Som(G) and whose set of arrows consists of the $f\in$ Fl(G) such that $\varphi (f)$ is an identity element of $G'$. For every $a\in$ Som(G)$,e_a\in H_{a,a}$. For every $f\in$ Fl(H), $\varphi (f^{-1}) =\varphi (f)^{-1}$, hence $f^{-1}\in$ Fl(H). Further, if $f$ and $g$ are composable arrows of H, one has $\varphi (f g) =\varphi (f)\varphi (g) =e_{t(\varphi(f))}e_{o(\varphi(g))}=$ $e_{t(\varphi(f))}$, hence $f g$ is an arrow of H. This shows that H is a subgroupoid of G. It is called the kernel of $\varphi$ and is denoted by Ker($\varphi$ ).