= Solution
For a <locally small category> $\mathcal C$, a <representation of a functor> $F:\mathcal C\to\mathbf{Set}$ is an object $A$ and an element $a\in F(A)$ such that
$$
\theta_B:\mathcal C(A,B)\longrightarrow F(B),
\qquad
f\longmapsto F(f)(a)
$$
is a bijection for every $B$, naturally in $B$. Equivalently, $\theta:\mathcal C(A,-)\cong F$ is a <natural transformation>[natural isomorphism].
Suppose $(A,a)$ and $(A',a')$ are two representations. Universality gives unique maps
$$
u:A\to A',\qquad v:A'\to A
$$
such that $F(u)(a)=a'$ and $F(v)(a')=a$. Then $F(vu)(a)=a$, and uniqueness applied to the element $a$ gives $vu=1_A$. Similarly $uv=1_{A'}$. Thus the representing objects are uniquely isomorphic in a way carrying one <universal element of a set-valued functor> to the other.
For $F:\mathcal C\to\mathcal D$ and $B\in\mathcal D$, the <comma category> $(B\downarrow F)$ has objects $(A,u)$ with $u:B\to F(A)$. A morphism $(A,u)\to(A',u')$ is a map $h:A\to A'$ satisfying
$$
F(h)u=u'.
$$
The <universal arrow from an object to a functor> criterion says that $F$ has a left adjoint exactly when $(B\downarrow F)$ has an <initial object> for every $B$. Indeed, an initial $(L B,\eta_B)$ represents the functor $A\mapsto\mathcal D(B,F A)$, and uniqueness makes $B\mapsto L B$ functorial.
When $F:\mathcal C\to\mathbf{Set}$ and $1$ is a singleton, an arrow $1\to F(A)$ is just an element $a\in F(A)$. Hence $(1\downarrow F)$ is the category of elements, and its initial objects are exactly the representations of $F$. This proves
$$
F\text{ representable}
\quad\Longleftrightarrow\quad
(1\downarrow F)\text{ has an initial object}.
$$
If $F$ has a left adjoint, the universal-arrow criterion immediately makes it representable. Conversely, suppose $\mathcal C$ is <cocomplete category>[cocomplete] and $F\cong\mathcal C(A,-)$. For a set $S$, form the copower
$$
L(S)=\coprod_{s\in S}A.
$$
The <coproduct in a category> universal property gives natural bijections
$$
\mathcal C(L(S),B)
\cong\prod_{s\in S}\mathcal C(A,B)
\cong\mathbf{Set}(S,F(B)),
$$
so $L\dashv F$.
Cocompleteness cannot be omitted. Let $\mathcal C$ be the <category of ordinals in reverse order>: there is one arrow $\alpha\to\beta$ exactly when $\alpha\geq\beta$ in the ordinary ordering. This large poset is locally small and complete. For a set-indexed family $(\alpha_i)$, its product in the reversed order is the ordinary supremum $\sup_i\alpha_i$, and equalizers in a poset are automatic. But $\mathcal C$ has no initial object, since that would be a largest ordinal. Any representable functor $\mathcal C(A,-)$ is therefore the requested example: if it had a left adjoint $L$, then
$$
\mathcal C(L(\varnothing),B)
\cong\mathbf{Set}(\varnothing,\mathcal C(A,B))
$$
would be a singleton for every $B$, making $L(\varnothing)$ initial, a contradiction.
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