Proposition 1

Let G be a groupoid. The relation "there exists an arrow of G joining $a$ to $b$" is an equivalence relation in the set of vertices of G whose equivalence classes are the orbits of G.

Let $a,b,c$ be vertices of G. One has $e_a\in G_{a,a}$; if $f\in G_{a,b}$, $f^{-1}\in G_{b,a}$; if $f\in G_{a,b}$ and $g\in G_{b,c}$, one has $f g\in G_{a,c}$. This shows that the indicated relation is reflexive, symmetric and transitive; it is therefore an equivalence relation. The second assertion then results from the definition of the connected components of a quiver.