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

Every covariant representable functor $\mathcal{C}(c,-)$ preserves categorical limits, since a morphism out of $c$ into a limit is precisely a compatible cone. A filtered colimit of these functors is computed pointwise, and filtered colimits commute with finite limits in sets. Interchanging the finite limit and filtered colimit therefore proves that $F$ preserves finite limits, which is (i). This completes the cycle. It is also the flat functor and finite-limit preservation criterion for a finitely complete small category.