Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 25 / 2 / b

With a terminal object $1$, the pullback in a category of $A\to 1\leftarrow B$ is the binary product in a category $A\times B$. Repetition gives all nonempty finite products in a category, and $1$ supplies the empty one.

For $f,g:A\rightrightarrows B$, form the diagonal ${\Delta }_B:B\to B\times B$ and pull it back along $⟨f,g⟩:A\to B\times B$. Write the resulting maps as $e:E\to A$ and $h:E\to B$. The equation $⟨f,g⟩e={\Delta }_{B}h$ says $fe=ge=h$. A map $x:X\to A$ with $fx=gx$ pairs with $fx:X\to B$ to give a unique map into this pullback in a category. Thus $e$ is the equalizer of $f,g$.

We now have the finite products in a category and equalizers needed in part (a). A terminal object and pullbacks in a category therefore suffice for all finite limits.