Proposition 7

For every groupoid morphism $\varphi : G\rightarrow G'$ which maps every arrow belonging to F onto an identity element of $G'$, there exists a unique groupoid morphism $\varphi ': G/F\rightarrow G'$ such that $\varphi '\circ p=\varphi$.

The kernel of $\varphi$ is a normal subgroupoid of G and the set of its arrows contains F. By definition, H is the smallest normal subgroupoid of G whose set of arrows contains F; it is therefore contained in the kernel of $\varphi$. The proposition then results from II, p. 170, Prop. 3.

Let us denote by Γ the subquiver of G whose set of vertices is Som(G) and whose set of arrows is F. The orbits of H are identified with the connected components of the quiver Γ. By definition of the quotient of a groupoid by a normal subgroupoid, the set of vertices of $G/F$ is identified with the set of connected components of Γ; in other words, it is the quotient set of Som(G) by the finest equivalence relation such that $o(f)$ is equivalent to $t(f)$ for every arrow $f\in F$.

The mapping $p$ induces, by passing to quotients, a bijection of the set of orbits of G onto the set of orbits of $G/F$ (II, p. 170, remark 1).

The purpose of this No$^o$ is to calculate the homomorphisms induced by $p$ by passing to the isotropy groups, which amounts, according to prop. 2 (II, p. 170), to calculating the isotropy groups of the subgroupoid H.