Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 20 / 3 / ii

Let $1$ be a terminal object. The pullback in a category of $X\to 1\leftarrow Y$ has precisely the universal property of the binary product in a category $X\times Y$: every pair of morphisms to $X,Y$ is compatible over $1$. Repeated binary products in a category, together with $1$ as the empty product, give all finite products in a category.

For $f,g:X\rightrightarrows Y$, form the pullback in a category of the categorical diagonal ${\Delta }_Y:Y\to Y\times Y$ along $⟨f,g⟩:X\to Y\times Y$. Its projection $e:E\to X$ satisfies $fe=ge$. If $h:T\to X$ satisfies $fh=gh$, the pair $h,fh$ gives a unique morphism into this pullback in a category. Conversely any such morphism must project to that pair. Hence $e$ is the equalizer of $f,g$. The preceding construction now gives all finite limits.