Pullback as Limit

Theorem
Let $\mathbf C$ be a metacategory.

Let $f_1: A \to C$ and $f_2: B \to C$ be morphisms of $\mathbf C$.

Let their pullback:


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

P \ar[r]^*+{p_2} \ar[d]_*+{p_1} & A \ar[d]^*+{f_1}

\\ B \ar[r]_*+{f_2} & C }\end{xy}$

exist in $\mathbf C$.

Then $\left({P, p_1, p_2}\right)$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined by:


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

\save[]+<0em,-2em>*{\mathbf{J}:} \restore & & \cdot \ar[d]

\\ & \cdot \ar[r] & \cdot }\end{xy}$


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

\save[]+<0em,-2em>*{D:} \restore & & C_1 \ar[d]^*+{f_1}

\\ & C_2 \ar[r]_*+{f_2} & C }\end{xy}$