Pullback as Equalizer/Corollary

Corollary to Pullback as Equalizer

Let $\mathbf C$ be a metacategory.

Then $\mathbf C$ has all pullbacks.

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}

&

}\end{xy}$

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

$\blacksquare$