A coequalizer of a pair of maps f, g:X->Y in a category is a map c:Y->C such that 1.c°f = c°g, where ° denotes composition. 2. For any other map c' :Y->C' with the same property, there is exactly one map γ:C->C' such that c' = γ°c, i.e., one has the above commutative diagram. It can be shown that the coequalizer is an epimorphism and that, moreover, it is unique up to isomorphism.

cokernel | commutative diagram | equalizer