Page "Equivalence relation" Paragraph 57

* An equivalence relation ~ on X is the equivalence kernel of its surjective projection π: X → X /~.

Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain.

Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.