Quotient Mapping is Coequalizer

Theorem

Let $\mathbf{Set}$ be the category of sets.

Let $S$ be a Set.

Let $\RR \subseteq S \times S$ be an equivalence relation on $S$.

Let $r_1, r_2: \RR \to S$ be the projections corresponding to the inclusion mapping $\RR \hookrightarrow S \times S$.

Then $q$ is a coequalizer of $r_1$ and $r_2$ in $\mathbf{Set}$.

Let $f: S \to T$ be a mapping as in the following commutative diagram:

$\begin{xy}\xymatrix{

\RR 
 \ar[r]<2pt>^*{r_1}
 \ar[r]<-2pt>_*{r_2}

&

S
 \ar[r]^*{q}
 \ar[rd]_*{f}

&

S / \RR
 \ar@{.>}[d]^*{\bar f}

\\

& &

}\end{xy}$

This translates to, for $s_1, s_2 \in S$ with $s_1 \RR s_2$:

$\map {f \circ r_1} {s_1, s_2} = \map {f \circ r_2} {s_1, s_2}$

that is:

$\map f {s_1} = \map f {s_2}$

The commutativity of the diagram implies that we must define $\bar f: S / \RR \to T$ by:

$\map {\bar f} {\eqclass {s_1} \RR} = \map f {s_1}$

since $\map q {s_1} = \eqclass {s_1} \RR$.

The above condition precisely states that $\bar f$ is well-defined.

In conclusion, for any $f$ with $f \circ r_1 = f \circ r_2$, there is a unique $\bar f$ making the diagram commute.

That is, $q$ is a coequalizer of $r_1$ and $r_2$.

$\blacksquare$