Given any set 
,
the associated pair groupoid is the set 
with the maps 
and 
, and multiplication 
. The inverse is 
. The left and right identity elements for 
are 
and 
, as is readily checked.
Any equivalence relation defines a subgroupoid 
of the pair groupoid 
, with 
if and only if 
. The orbits of 
are then the equivalence classes.
Given any groupoid 
over 
,
the map 
is a morphism of groupoids.
