= Solution
Suppose $F\cong\operatorname*{colim}_{j\in J}R_j$, where $J$ is a <filtered category> and every $R_j$ is a <representable functor>. Every covariant <representable functor> $\mathcal C(A,-)$ preserves all existing <categorical limits>: maps from $A$ into a limiting object are exactly compatible families of maps into its diagram.
For a finite <diagram in a category> $K:I\to\mathcal C$, use pointwise <colimits> in the <functor category>, preservation by each $R_j$, and the permitted result that <filtered colimits commute with finite limits in sets> to obtain
$$
\begin{aligned}
F(\lim_iK_i)
&\cong\operatorname*{colim}_jR_j(\lim_iK_i)\\
&\cong\operatorname*{colim}_j\lim_iR_j(K_i)\\
&\cong\lim_i\operatorname*{colim}_jR_j(K_i)
\cong\lim_i F(K_i).
\end{aligned}
$$
The isomorphism is the canonical comparison, since every step respects the limiting projections. This also covers the empty finite diagram and hence the <terminal object>. Thus \b[$(iv)\Rightarrow(i)$], completing the cycle:
$$
\boxed{(i)\Longleftrightarrow(ii)\Longleftrightarrow(iii)\Longleftrightarrow(iv).}
$$
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