Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 17 / 4 / i / Solution

First construct every finite product in a category by iterating binary products in a category; the empty product is the terminal object. Let $D:J\to \mathcal{C}$ be a diagram in a category with finitely many objects and arrows. Form the finite products in a categoryDefine $a,b:P\rightrightarrows Q$ by their componentsTake their equalizer $e:L\to P$. Its components $p_j={\pi }_{j}e$ satisfy $D(u)p_i=p_j$, so they form a cone over a diagram.

For any other categorical cone $x_j:X\to D(j)$, the product in a category property supplies a unique $x:X\to P$ with ${\pi }_{j}x=x_j$. The cone equations imply $ax=bx$, so the equalizer property supplies a unique $\bar{x}:X\to L$ with $e\bar{x}=x$. These are exactly the required equations $p_j\bar{x}=x_j$, with uniqueness. Thus $L$ is a categorical limit. For empty $J$, both products are terminal and this construction still gives a terminal limit. All finite limits exist.