= Solution
Let $1$ be a <terminal object>. The <pullback in a category> of $X\to1\leftarrow Y$ 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 $\langle f,g\rangle:X\to Y\times Y$. 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 \b[all <finite limits>].
Back to article page
