Exercise 1, § 5

Soient G a groupoid, X a set and $f$ a mapping from Som(G) into X.

a) Prove that there exists a pair $(K, \alpha )$, where K is a groupoid with set of vertices X and $\alpha : G\rightarrow K$ is a morphism of groupoids such that Som($\alpha$ ) $=f$, satisfying the following universal property: for every groupoid H with set of vertices X and every morphism $\varphi : G\rightarrow H$ such that Som($\varphi$ ) $=f$, there exists a unique morphism of groupoids $\psi : K\rightarrow H$ such that $\alpha \circ \psi =\varphi$. (If S is a set, we denote by $S_d$ a groupoid with set of vertices S whose orbits and isotropy groups are reduced to one element. Let $G'$ be the disjoint sum groupoid of the family $(G,X_d)$, let $\varphi$ and $\psi$ be the morphisms of the groupoid Som(G)$_d$ into $G'$ such that Som($\varphi$ ) is the canonical injection of Som(G) into Som(G$'$) and Som($\psi$ ) $=f$ respectively. Put K = Coeg($\varphi , \psi$ ).)

b) Deduce a bijection of Hom(G$, f^*H$) onto Hom(K$,H$), for every groupoid H with set of vertices X.