We claim that $A\times _{C}B$ is the categorical pullback of $A$ and $B$ over $C$ with respect to $\webleft (f,g\webright )$ in $\mathsf{Sets}$. First we need to check that the relevant pullback diagram commutes, i.e. that we have
Indeed, given $\webleft (a,b\webright )\in A\times _{C}B$, we have
\begin{align*} \webleft [f\circ \text{pr}_{1}\webright ]\webleft (a,b\webright ) & = f\webleft (\text{pr}_{1}\webleft (a,b\webright )\webright )\\ & = f\webleft (a\webright )\\ & = g\webleft (b\webright )\\ & = g\webleft (\text{pr}_{2}\webleft (a,b\webright )\webright )\\ & = \webleft [g\circ \text{pr}_{2}\webright ]\webleft (a,b\webright ),\end{align*}
where $f\webleft (a\webright )=g\webleft (b\webright )$ since $\webleft (a,b\webright )\in A\times _{C}B$. Next, we prove that $A\times _{C}B$ satisfies the universal property of the pullback. Suppose we have a diagram of the form
in $\mathsf{Sets}$. Then there exists a unique map $\phi \colon P\to A\times _{C}B$ making the diagram
commute, being uniquely determined by the conditions
\begin{align*} \text{pr}_{1}\circ \phi & = p_{1},\\ \text{pr}_{2}\circ \phi & = p_{2}\end{align*}
via
\[ \phi \webleft (x\webright )=\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright ) \]
for each $x\in P$, where we note that $\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright )\in A\times B$ indeed lies in $A\times _{C}B$ by the condition
\[ f\circ p_{1}=g\circ p_{2}, \]
which gives
\[ f\webleft (p_{1}\webleft (x\webright )\webright )=g\webleft (p_{2}\webleft (x\webright )\webright ) \]
for each $x\in P$, so that $\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright )\in A\times _{C}B$.
