Universal property of the quotient topology

If $q:X\to X/R$ is the quotient map and a continuous map $f:X\to Y$ is constant on each equivalence class, there is a unique continuous map $F:X/R\to Y$ satisfying $Fq=f$. It is given by $F([x])=f(x)$; continuity follows becausefor every open $U\subseteq Y$.