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

Take $A=1$. The category of elements $\int F=(1\downarrow F)$ has finite limits. Reversing its limiting cones gives cocones over every finite diagram in $(\int F{)}^{\mathrm{op}}$; a terminal object before reversal ensures nonemptiness. Its opposite category is therefore a filtered category, proving (iii).