= Solution
Take $A=1$. By assumption $E=(1\downarrow F)$ has <finite limits>, so $E^{\mathrm{op}}$ has finite <colimits>. A <filtered category> is nonempty, admits a cocone over every pair of objects, and admits a coequalizing arrow for every parallel pair. The <initial object> of $E^{\mathrm{op}}$ gives nonemptiness, its binary <coproducts> give the required two-object cocones, and its <coequalizers> equalize parallel arrows. Consequently $\boxed{\text{(ii)}\Rightarrow\text{(iii)}}$. These statements also handle the otherwise possible empty <category of elements>.
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