Pullback as Equalizer/Corollary

Corollary to Pullback as Equalizer
Let $\mathbf C$ be a metacategory. Suppose $\mathbf C$ has all binary products and all equalizers.

Then $\mathbf C$ has all pullbacks.

Proof
Let $f: A \to C$ and $g: B \to C$ be morphisms of $\mathbf C$ with common codomain.

By assumption on $\mathbf C$, there exists a binary product $A \times B$ with projections $\pi_1: A \times B \to A$ and $\pi_2: A \times B \to B$.

Again by assumption on $\mathbf C$, there exists an equalizer $e: P \to A \times B$ of $f \circ \pi_1$ and $g \circ \pi_2$.

From Pullback as Equalizer, the pullback of $f$ and $g$ is given by:


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

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

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

where $p_1 = \pi_1 \circ e$ and $p_2 = \pi_2 \circ e$.