In mathematics, given a category C, a quotient of an object X by an equivalence relation is a coequalizer for the pair of maps where R is an object in C and "f is an equivalence relation" means that, for any object T in C, the image (which is a set) of is an equivalence relation; that is, a reflexive, symmetric and transitive relation. The basic case in practice is when C is the category of all schemes over some scheme S. But the notion is flexible and one can also take C to be the category of sheaves.