Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 1 / b / Solution

Let $1$ be a terminal object. The pullback of the unique arrows $A\to 1\leftarrow B$ is a binary categorical product: maps $X\to A$ and $X\to B$ automatically have equal composites into $1$, so the pullback universal property is exactly the categorical product universal property.

For parallel arrows $f,g:A\rightrightarrows B$, form $B\times B$ and take the pullback of $⟨f,g⟩:A\to B\times B$ along the diagonal $\delta :B\to B\times B$. Write the resulting projections as $e:E\to A$ and $b:E\to B$. The pullback equation givesIf $u:X\to A$ satisfies $fu=gu$, the pair $(u,fu)$ is a pullback cone, producing a unique $v:X\to E$ with $ev=u$ and $bv=fu$. The second condition is forced by the first, since $b=fe$. Thus $e$ is the equaliser of $f,g$.

The terminal object and iterated binary categorical products give all finite categorical products, including the nullary one. Part (a) therefore proves pullbacks and a terminal object give all finite categorical limits.