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

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## 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**.