= Solution
If $G$ is a <representable functor>, it preserves small <categorical limits>. Part (b) makes $(1\downarrow G)$ have an <initial object>, which gives a <weakly initial set>.
Conversely, assume limit preservation and a <weakly initial set> in $(1\downarrow G)$. The needed comma-limit fact is explicit: for a small <diagram in a category> $(A_j,a_j)$, form $L=\lim_j A_j$ in $\mathcal C$. The elements $a_j\in G(A_j)$ form a compatible family, hence determine a unique $a\in G(L)$ under $G(L)\cong\lim_jG(A_j)$. The projections from $(L,a)$ are a <categorical limit> in the <comma category>. This includes the empty diagram, since $G$ carries the <terminal object> to a singleton. Each comma <hom-set> is a subset of a <hom-set> in $\mathcal C$, so the <comma category> is also <locally small>.
Part (c) now gives an <initial object> of $(1\downarrow G)$, and part (b) supplies a <representation of a functor>. Therefore
$$
\boxed{G\text{ is representable}\iff G\text{ preserves small limits and }(1\downarrow G)\text{ has a weakly initial set}.}
$$
The size requirement is a genuine <solution-set condition>; limit preservation alone does not produce it in an arbitrary <complete category>.
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