Quotient by an equivalence relation

In mathematics, given a category "C", a quotient of an object "X" by an equivalence relation formula_1 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 formula_3 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.