Example 4

Let G be a groupoid, let X be a set and let $\varphi : X\rightarrow$ Som(G) be a mapping. Define a graph $G'$ in the following way. The set Som(G$'$) is the set X; for every pair $(x, y)$ of elements of X, the set Fl$_{x,y}(G')$ is the set of triples $(x, f, y)\in X\times$ Fl(G) $\times X$, where $f$ is an element of Fl$_{\varphi(x),\varphi(y)}(G)$. Let $x,y,z$ be elements of X$,f\in$ Fl$_{\varphi(x),\varphi(y)}(G)$ and $g\in$ Fl$_{\varphi(y),\varphi(z)}(G)$ arrows of G, and put $(x, f, y)\cdot (y, g, z) = (x, f g, z)$. This defines a composition law in the graph $G'$ which makes it a groupoid. It is called the inverse image groupoid of the groupoid G by the mapping $\varphi$ and is denoted by $\varphi^*(G)$.

The pair $(\varphi , \psi )$, where $\psi :$ Fl($\varphi^*(G)$)$\rightarrow$ Fl(G) is the mapping defined by $(x, f, y)\mapsto f$ is a groupoid morphism from $\varphi^*(G)$ into G, called the canonical morphism.