= Solution
Let $u_{C,x}:C\to R$ be a <colimit> cocone for $G:\operatorname{Elts}(X)\to\mathcal C$. Here the preservation hypothesis must include the <categorical limit> of $G^{\mathrm{op}}$ in $\mathcal C^{\mathrm{op}}$; this indexing <category> can be large. For a <morphism> $f:(C,x)\to(D,y)$ of the <category of elements>, the defining equation $X(f)(y)=x$ says that the family of distinguished elements $(x)_{(C,x)}$ is compatible. Preservation gives a unique $r\in X(R)$ with
$$
X(u_{C,x})(r)=x\quad\text{for every }(C,x).
$$
In particular each $u_{C,x}$ is now a <morphism> $(C,x)\to(R,r)$ in the <category of elements>. Naturality of the <colimit> cocone gives
$$
u_{R,r}\,u_{C,x}=u_{C,x}.
$$
The <universal property> of the <colimit> forces $u_{R,r}=1_R$, since it and $1_R$ agree after every cocone leg. If $f:C\to R$ also satisfies $X(f)(r)=x$, it is a <morphism> $(C,x)\to(R,r)$, and cocone compatibility now yields $u_{C,x}=u_{R,r}f=f$. Thus $(R,r)$ is a <universal element>. By the preceding representability criterion, \b[the presheaf is represented by the colimit object]:
$$
\boxed{\mathcal C(-,R)\cong X,\qquad f\longmapsto X(f)(r).}
$$
The size qualification matters. If preservation means only small <categorical limits>, the implication as printed is false for general <locally small categories>. For an explicit counterexample, let $\mathcal C$ be the ordered <category> of all <ordinals> with an extra greatest object $\infty$. Put $X(\alpha)=\{*\}$ for every <ordinal> $\alpha$ and $X(\infty)=\varnothing$, with the forced restriction maps. Every small <colimit> in $\mathcal C$ is the supremum of its object values: it is an <ordinal> unless the diagram contains $\infty$. Applying $X$ therefore gives the appropriate small <categorical limit> of singletons, or the empty <set> when a value is empty; an empty diagram gives $X(0)=\{*\}$. Hence $X$ preserves all small <categorical limits>.
Its <category of elements> consists of all <ordinals>, whose projection has the large <colimit> $\infty$ in $\mathcal C$. Nevertheless, no <ordinal> represents $X$, because its representable vanishes on larger ordinals; $\infty$ does not represent it either, since $\mathcal C(\infty,\infty)$ is nonempty while $X(\infty)$ is empty. The valid proof consequently uses preservation of the displayed possibly large limit, or a smallness hypothesis making that diagram small.
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