Corollary

Let $X, Y$ be two topological spaces, $R$ (resp. $S$) an equivalence relation on $X$ (resp. $Y$), and let $f : X \to Y$ be a continuous mapping which is compatible with the equivalence relations $R$ and $S$ (Set Theory R, § 5, no. 8); then the mapping $g : X/R \to Y/S$ induced by $f$ (Set Theory R, § 5, no. 8) is continuous.

This is a particular case of a general property of quotient structures (Set Theory, Chapter IV, § 2, no. 6, criterion CST 20).