\begin{definition}[Definition:Equalizer]
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 the limit of the diagram $C \underset f{\overset g\rightrightarrows} D$, that 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:
:::$\begin{xy}\xymatrix{
 E 
  \ar[r]^*{e}
&
 C
  \ar[r]<2pt>^*{f}
  \ar[r]<-2pt>_*{g} 
&
 D
\\
 A
  \ar@{.>}[u]^*{u}
  \ar[ur]_*{a}
}\end{xy}$
:is a commutative diagram. I.e., $a = e \circ u$.
\end{definition}