= Solution
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 $L=\lim_jA_j$ in $\mathcal C$. Its distinguished elements form a compatible family in $\lim_jGA_j$. <Categorical limit> preservation gives a unique $x\in GL$ with $Gp_j(x)=x_j$. The underlying <categorical limit> factorization of a <categorical cone> preserves this element, proving the comma-category <universal property>. For the empty diagram, this uses $G1\cong1$. 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.
Back to article page
