= 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
$$
\boxed{(ii)\Rightarrow(iii):\quad(1\downarrow F)^{\mathrm{op}}\text{ is filtered}.}
$$
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