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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., $a = e \circ u$.

Also see