Definition:Equalizer

Definition
Let $\mathbf C$ be a metacategory.

Let $f, g: C \to D$ be morphisms with common domain and codomain.

An equalizer for $f$ and $g$ is a morphism $e: E \to C$ such that:


 * $f \circ e = g \circ e$

and subject to the following UMP:


 * For any $a: A \to C$ such that $f \circ a = g \circ a$, there is a unique $u: A \to E$ such that:


 * $\begin{xy}\xymatrix{

E  \ar[r]^*{e} & C \ar[r] ^*{f} \ar[r]<-2pt>_*{g} & D

\\ A \ar@{.>}[u]^*{u} \ar[ur]_*{a} }\end{xy}$


 * is a commutative diagram. I.e., $a = e \circ u$.

Also see

 * Coequalizer, the dual notion