Pullback as Equalizer

Theorem
Let $\mathbf C$ be a metacategory with binary products.

Suppose that the following diagram commutes (possibly except for the square):


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

P \ar@/^/[rrd]^*+{p_1} \ar@/_/[ddr]_*+{p_2} \ar[rd]^*+{e}

\\ & A \times B \ar[r]_*+{\pi_1} \ar[d]^*+{\pi_2} & A \ar[d]^*+{f}

\\ & B \ar[r]_*+{g} & C }\end{xy}$

where $A \times B$ is the product of $A$ and $B$, and $\pi_1, \pi_2$ are the associated projections.

Then the border of the diagram, i.e.:


 * $\begin{xy}\xymatrix{

P \ar[r]^*+{p_1} \ar[d]_*+{p_2} & A \ar[d]^*+{f}

\\ B \ar[r]_*+{g} & C }\end{xy}$

is a pullback of $f$ and $g$ iff $e: P \to A \times B$ is the equalizer of $f \circ \pi_1$ and $g \circ \pi_2$.