Proposition 5

Let $f : G \to G'$ be a homomorphism of groups with operators and $H$ and $H'$ normal stable subgroups of $G$ and $G'$ respectively such that $f(H) \subset H'$. The mapping $f$ is compatible with the equivalence relations defined by $H$ and $H'$. Let $\pi : G \to G/H$ and $\pi' : G' \to G'/H'$ be the canonical homomorphisms. The mapping $\bar{f} : G/H \to G'/H'$ derived from $f$ by passing to the quotients is a homomorphism.

If $x \equiv y \pmod{H}$, then $x^{-1}y \in H$, whence

and hence $f(x) \equiv f(y) \pmod{H'}$. The second assertion follows from the universal property of quotient laws ($§ 1$, no. 6).