An equalizer is a limit of a diagram indexed by the parallel pair, the category with two objects and two parallel non-identity morphisms. A diagram of this shape is simply a parallel pair of morphisms in the target category . A cone over this diagram with summit consists of a pair of morphisms and so that and ; these two equations correspond to the naturality conditions \eqref{eq:cone-naturality} imposed by each of the two non-identity morphisms in the indexing category. Together, they assert that ; the morphism is necessarily equal to this common composite. Thus, a cone over a parallel pair is represented by a single morphism so that .
The equalizer is the universal arrow with this property. The limit diagram \(\xymatrix{ E \ar[r]^h & A \ar@<.5ex>[r]^f \ar@<-.5ex>[r]_g & B}\) is called an equalizer diagram. The universal property asserts that given any so that , there exists a unique factorization of through . \(\xymatrix{ C \ar[dr]^a \ar@{-->}[d]^{\exists !}_k \\ E \ar[r]_h & A \ar@<.5ex>[r]^f \ar@<-.5ex>[r]_g & B}\)