What is Quotient set?

Let $\mathcal{R}$ be an equivalence relation on $E$. The quotient set $E/\mathcal{R}$ is the set of equivalence classes; the surjection $\pi \colon E \to E/\mathcal{R}$, $x \mapsto \mathrm{cl}(x)$, is the canonical projection.

Universal property (factorization): if $f \colon E \to F$ is compatible with $\mathcal{R}$ (i.e. $x \mathbin{\mathcal{R}} y \implies f(x) = f(y)$), there is exactly one map $\overline f \colon E/\mathcal{R} \to F$ with $f = \overline f \circ \pi$.