Remark 2

The mapping Orb($\gamma$ ), deduced from $\gamma$ by passing to orbits, defines by passage to the quotient a bijection of the set of connected components of the framework of the pair $(\varphi , \psi )$ onto the set of orbits of the coequalizer. This results from II, p. 185, prop. 4 and from the fact that the mapping deduced from $\beta$ by passing to orbits is bijective ( II, p. 170, remark 1).

Consequently, the mapping of Som(G) into Orb(Coeg($\varphi , \psi$ )) deduced from $\gamma$ identifies the set of orbits of Coeg($\varphi , \psi$ ) with the quotient set of Som(G) by the equivalence relation generated by the pairs $(\varphi (x), \psi (x))$ for $x\in$ Som(H) and the pairs $(o(f), t(f))$ for $f\in$ Fl(G).