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

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.

Use the explicit construction of small limits from products and equalizers from part (i), here with finite indexing sets. If $F$ preserves finite products in a category and equalizers, then $F(P)$ and $F(Q)$, with their image projections, are the products of the objects $FD(j)$ and of the target objects indexed by arrows of $J$.

The image maps $F(a),F(b)$ have componentsAlso $F(e):F(L)\to F(P)$ is their equalizer. Therefore the same universal property proof makes $F(L)$, with projections $F(p_j)$, a categorical limit of $FD$. Any other chosen limit of $D$ is uniquely isomorphic to this construction, and the functor carries that isomorphism to an isomorphism. Consequently $F$ preserves every finite limit. Preservation of finite products includes the empty product, so the terminal object is also preserved.

Given $A,B$, their product in a category is the pullback of the unique arrows $A\to 1\leftarrow B$, where $1$ is a terminal object. Having constructed this product in a category, define the equalizer of $f,g:A\rightrightarrows B$ as the pullbackwhere ${\delta }_B$ is the categorical diagonal. A map $x:X\to A$ factors through this pullback exactly when $fx=gx$: the accompanying arrow to $B$ must equal this common composite. The factorization is unique, proving the equalizer property.

Thus the hypotheses give a terminal object, binary products in a category and all equalizers. Part (i) now gives all finite limits.