Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 23 / 5 / i / Solution

Assume $F$ preserves finite limits. Given any finite diagram in the comma category $(A\downarrow F)$, first take its underlying limit $c$ in $\mathcal{C}$. Its structure maps $A\to F{c}_j$ form a cone, and preservation gives a unique map $A\to Fc$. This object and the underlying projections satisfy the limiting universal property in the comma category. The construction also includes the empty diagram, since $F$ sends a terminal object to a singleton. Thus $(A\downarrow F)$ has finite limits for every set $A$, proving (ii).