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If $G$ is representable, it is a limit-preserving functor, and its universal element gives an initial object of $(1\downarrow G)$, hence a weakly initial singleton.

Conversely, a limit-preserving $G$ makes $(1\downarrow G)$ a complete category. For a small diagram $(A_j,x_j)$, take $L={\lim }_j{A}_j$ in $\mathcal{C}$. Its distinguished elements form a compatible family in ${\lim }_{j}G{A}_j$. Categorical limit preservation gives a unique $x\in GL$ with $G{p}_j(x)=x_j$. The underlying categorical limit factorization of a categorical cone preserves this element, proving the comma-category universal property. For the empty diagram, this uses $G1\cong 1$. This is the construction of limits in a comma category of a limit-preserving functor.

The comma category is locally small since its arrows are subsets of the hom-sets in $\mathcal{C}$. By the previous part, its weakly initial set therefore yields an initial object. Part (b) then yields a representation of $G$. Thus representability from a solution set follows with all small categorical limits, rather than finite categorical limits alone.