Example 1

Let X be a set; let us denote by $X\times X$ the groupoid of pairs of X (II, p. 163, Example 2). Let R be an equivalence relation on X. The subquiver of the groupoid $X\times X$ whose set of vertices is X and whose set of arrows is the graph of the equivalence relation R is a subgroupoid of $X\times X$. Its orbits are the equivalence classes of the relation R.

Conversely, every subgroupoid of $X\times X$ whose set of vertices is X is of this form.

Let X and S be sets and let $p: X\rightarrow$ S be a mapping. The mapping $p$ defines an equivalence relation in X (E, II, p. 41). One denotes by $X\times_SX$ the subgroupoid of $X\times X$ defined by this equivalence relation, and one calls it the groupoid of pairs of $(X, p)$.