= Solution
First address the unnumbered preliminaries. The \b[<Yoneda lemma>] gives a <natural transformation> bijection
$$
\boxed{\operatorname{Nat}(\mathcal C(A,-),F)\cong F(A),\qquad\tau\longmapsto\tau_A(1_A).}
$$
The element $x\in F(A)$ corresponds to $\tau_B(f)=F(f)(x)$ for $f:A\to B$. These bijections are natural in $A$ and $F$; the contravariant form replaces $\mathcal C(A,-)$ by $\mathcal C(-,A)$.
Let $E=(1\downarrow F)$ be the <category of elements>. Its objects are $(A,x)$ with $x\in F(A)$; an arrow $u:(A,x)\to(B,y)$ is $u:A\to B$ with $F(u)(x)=y$. Since $\mathcal C$ is a <small category>, $E$ is small. Define a <diagram in a category> $D:E^{\mathrm{op}}\to[\mathcal C,\mathbf{Set}]$ by
$$
D(A,x)=\mathcal C(A,-).
$$
For the opposite of $u$, the map $\mathcal C(B,-)\to\mathcal C(A,-)$ is precomposition with $u$. The <Yoneda lemma> gives a <cocone under a diagram> $D\to F$ whose $(A,x)$ component sends $f:A\to B$ to $F(f)(x)$.
For any $G$, a competing <cocone under a diagram> amounts, by the <Yoneda lemma>, to elements $z_{A,x}\in G(A)$ satisfying
$$
G(u)(z_{A,x})=z_{B,F(u)(x)}.
$$
This is exactly the <naturality> condition for $\alpha_A:F(A)\to G(A)$ defined by $\alpha_A(x)=z_{A,x}$. It gives a unique <natural transformation> $\alpha:F\to G$ factoring the cocone. Thus the <universal property> of a <colimit> proves the <canonical colimit presentation of a covariant set-valued functor>:
$$
\boxed{F\cong\operatorname*{colim}_{(A,x)\in E^{\mathrm{op}}}\mathcal C(A,-).}
$$
The opposite on $E$ is essential for the variance of the <representable functors>.
We now prove the four conditions equivalent by the cycle $(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(iv)\Rightarrow(i)$. Suppose $F$ is a <finite-limit-preserving set-valued functor>. For a finite <diagram in a category> in the <comma category> $(S\downarrow F)$, write its objects as $(A_i,s_i:S\to F(A_i))$. Take $L=\lim_i A_i$ in $\mathcal C$. Since $F$ preserves this <finite limit>, the compatible maps $s_i$ determine a unique $s:S\to F(L)$ whose composites with $F(L)\to F(A_i)$ are $s_i$. The pair $(L,s)$ has the required <categorical limit> property in $(S\downarrow F)$, by the same universal properties. For the empty diagram, $F(1)$ is a singleton, so there is exactly one map $S\to F(1)$; this provides the <terminal object>. Hence \b[$(i)\Rightarrow(ii)$ for every set $S$].
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