Equivalence relations on sets

Suppose $E$ is an equivalence relation on a set $X$ . Show that the usual set $X / E$ of equivalence classes can be described by a coequalizer in $Set$ .

Hints

Recall that an equivalence relation on $X$ is a subset $E \subseteq X \times X$ satisfying various properties (e.g., transitive, etc.). Two elements $x_{1}, x_{2} \in X$ are said to be equivalent exactly when $(x_{1}, x_{2}) \in E$ . As such, the set $E$ comes with two projection maps to $X$ . These are your parallel arrows that the quotient set $X / E$ will "coequalize." Speaking of which, the set $X / E$ is defined to be the collection of equivalence classes in $X$ . Each element in $X / E$ is a subset $S \subseteq X$ consisting of all elements equivalent to each other. Each such subset $S$ is usually denoted $[x]$ , where $x \in X$ is any representative of that subset. In other words, we have $[x_{1}] = [x_{2}]$ exactly when $(x_{1}, x_{2}) \in E$ .

The quotient set $X / E$ also comes with a projection map $π : X \to X / E$ . This is what is claimed to be the coequalizer of your parallel arrows.