= Solution
For the <categorical presheaf> $X$, its <category of elements> has objects $(C,x)$ with $x\in X(C)$. A <morphism> $(C,x)\to(D,y)$ is a <morphism> $f:C\to D$ satisfying $X(f)(y)=x$. <Composition in a category> is inherited from $\mathcal C$: if also $X(g)(z)=y$, then $X(gf)(z)=X(f)X(g)(z)=x$. The <identity morphisms> are inherited as well. The <forgetful functor> sends $(C,x)$ to $C$ and $f$ to $f$.
A <universal element> is a pair $(R,r)$ for which each $x\in X(C)$ is uniquely of the form $X(f)(r)$ for $f:C\to R$. Thus $(R,r)$ is a <terminal object> of the <category of elements>, with the variance appropriate to a <categorical presheaf>.
Given a <universal element>, define
$$
\psi_C:\mathcal C(C,R)\longrightarrow X(C),\qquad f\longmapsto X(f)(r).
$$
The defining uniqueness makes each map a <bijection>; $X(u)\psi_C(f)=\psi_{C'}(fu)$ gives <naturality> for $u:C'\to C$. Hence $\psi$ is a <natural isomorphism> and $X$ is a <representable presheaf>. Conversely, from a <natural isomorphism> $\psi:\mathcal C(-,R)\cong X$, take $r=\psi_R(1_R)$. The <Yoneda lemma> gives $\psi_C(f)=X(f)(r)$; its bijectivity makes $(R,r)$ a <universal element>. Therefore \b[the two descriptions coincide]:
$$
\boxed{X\text{ is representable}\iff\operatorname{Elts}(X)\text{ has a terminal object}.}
$$
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