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 $\struct {E, e}$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined by:


 * $\begin{xy}\xymatrix@+1em@L+3px{

\mathbf{J}: & \ast \ar[r] \ar[r]<-2pt> & \star }\end{xy}$


 * $\begin{xy}\xymatrix@+1em@L+3px{

D: & C_1 \ar[r] ^*+{f_1} \ar[r]<-2pt>_*+{f_2} & C_2 }\end{xy}$