Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 119 / 2 / ii / Solution

Apply condition $(ii)$ with the singleton $S=1$. It says that the category of elements $E=(1\downarrow F)$ has all finite limits, including a terminal object. The opposite category $E^{\mathrm{op}}$ therefore has all finite colimits. In particular it is nonempty; two objects map to their binary coproduct in a category; and any parallel arrows are equalized by their coequalizer. These are precisely the conditions for a filtered category. Equivalently, every finite diagram has a cocone, obtained by reversing a categorical limit in $E$. Thus