# Equalizer as Limit

## Theorem

Let $\mathbf C$ be a metacategory.

Let $f_1, f_2: C_1 \to C_2$ be morphisms of $\mathbf C$.

Let their equalizer $e: E \to C_1$ exist in $\mathbf C$.

Then $\left({E, e}\right)$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined by:

$\begin{xy}\[email protected][email protected]+3px{ \mathbf{J}: & \ast \ar[r]<2pt> \ar[r]<-2pt> & \star }\end{xy}$
$\begin{xy}\[email protected][email protected]+3px{ D: & C_1 \ar[r]<2pt>^*+{f_1} \ar[r]<-2pt>_*+{f_2} & C_2 }\end{xy}$