An equalizer is a limit of a diagram indexed by the parallel pair, the category $\bullet \rightrightarrows \bullet$ with two objects and two parallel non-identity morphisms. A diagram of this shape is simply a parallel pair of morphisms $f,g \colon A \rightrightarrows B$ in the target category $\mathsf{C}$. A cone over this diagram with summit $C$ consists of a pair of morphisms $a \colon C \to A$ and $b \colon C \to B$ so that $fa = b$ and $ga = b$; 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 $fa = ga$; the morphism $b$ is necessarily equal to this common composite. Thus, a cone over a parallel pair $f,g \colon A \rightrightarrows B$ is represented by a single morphism $a \colon C \to A$ so that $fa = ga$.
The equalizer $h \colon E \to A$ 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 $a \colon C \to A$ so that $fa = ga$, there exists a unique factorization $k \colon C \to E$ of $a$ through $h$. \(\xymatrix{ C \ar[dr]^a \ar@{-->}[d]^{\exists !}_k \\ E \ar[r]_h & A \ar@<.5ex>[r]^f \ar@<-.5ex>[r]_g & B}\)