Proposition 2

Let $a$ be a vertex of G. The group homomorphism $p_a: G_a\rightarrow (G/H)_{p(a)}$ deduced from $p$ by passing to the isotropy groups is surjective; its kernel is $H_a$.

The homomorphism $p_a$ is surjective by virtue of the preceding remark. Its kernel contains $H_a$ by construction of the groupoid $G/H$. Let $f$ be an element of $G_a$ such that $p_a(f) =e_{p(a)}$. This means that $f$ and $e_a$ are equivalent modulo $\mathscr{R}$; there then exist arrows $x$ and $y$ of H such that $f=xe_ay$. Necessarily, $x$ and $y$ belong to $H_a$, whence $f\in H_a$.