# Definition:User:Dfeuer/Renaming Mapping/Definition 2

## Definition

Let $S, T$ be sets.

Let $\mathcal R$ be an equivalence relation on $S$.

Let $f: S \to T$ be any mapping.

Let $q_\mathcal R: S \to S / \mathcal R$ be the quotient mapping induced by $\mathcal R$.

Suppose that there is a mapping $r: S / \mathcal R \to T$ such that:

$r \circ q_\mathcal R = f$

Then $r$ is called the renaming mapping.