= Solution
It is enough to prove the <solution-set condition> and then apply the <general adjoint functor theorem>. Fix $D\in\mathcal D$ and a <morphism> $d:D\to UC$. We will factor it through one object in a <set> depending only on $D$.
First construct a <minimal supported subobject>. Among the <subobjects> $m:M\hookrightarrow C$ for which $d=U(m)d_M$ for some $d_M:D\to UM$, include $1_C$ and take their intersection $m_0:C_0\hookrightarrow C$. This is a small intersection, by well-poweredness. It exists by completeness as the <categorical limit> of the diagram consisting of these <monomorphisms> into $C$. Its map to $C$ is a <monomorphism>: two maps with the same composite to $C$ have equal projections to every $M$, since each $m$ is monic, and are then equal by the <categorical limit> property. One can equivalently construct these intersections by <pullbacks in a category> and small <products in a category>, using stability of <monomorphisms> under <pullback in a category>.
Choose the factorizations $d_M$. They form a compatible <categorical cone> into the image diagram under $U$, all with common composite $d$ to $UC$. Preservation of small <categorical limits> yields $d_0:D\to UC_0$ with $U(m_0)d_0=d$. If $n:N\hookrightarrow C_0$ is another <subobject> through which $d_0$ factors after applying $U$, then $m_0n$ is among the original supported <subobjects>. The intersection property gives $r:C_0\to N$ with $m_0nr=m_0$. Since $m_0$ is monic, $nr=1_{C_0}$; since $n$ is monic, also $rn=1_N$. Thus \b[every supported subobject of $C_0$ is invertible].
For each member $Q_i$ of the <small cogenerating family>, consider
$$
\theta_i:\mathcal C(C_0,Q_i)\longrightarrow\mathcal D(D,UQ_i),\qquad q\longmapsto U(q)d_0.
$$
This map is <injective>. If $U(q)d_0=U(q')d_0$, preservation of the <equalizer> of $q,q'$ makes $d_0$ factor through the image of that <equalizer>. Minimality makes its inclusion an <isomorphism>, so $q=q'$.
Write $R_i=\mathcal D(D,UQ_i)$; it is a <set> by local smallness of $\mathcal D$. Let $S_i\subseteq R_i$ be the image of $\theta_i$. For each $s\in S_i$, there is exactly one corresponding $q_{i,s}:C_0\to Q_i$. These maps define the <evaluation embedding into cogenerator products>
$$
e:C_0\longrightarrow P_S:=\prod_{i\in I}\prod_{s\in S_i}Q_i.
$$
It is a <monomorphism>: if $eg=eh$ and $g\ne h$, cogeneration supplies some $q:C_0\to Q_i$ distinguishing $g,h$; that $q$ is one of the projections of $e$, a contradiction. The product is small. It is important to use the subfamilies $S_i$, since some missing coordinate in $R_i$ need not correspond to a <morphism> out of $C_0$.
There are only a <set> of possible families $(S_i\subseteq R_i)_{i\in I}$. For each such family form $P_S$, choose a <set> of representatives $N\hookrightarrow P_S$ of its <subobjects>, and take all pairs $(N,d_N:D\to UN)$. Local smallness of $\mathcal D$ and well-poweredness make their union a <set>. In the case just constructed, $e$ identifies $C_0$ with one chosen representative $N$. Transporting $d_0$ to $UN$ and composing the inverse identification with $m_0$ gives a factorization of the original $d$ through that pair. Therefore these pairs form a <weakly initial set> in $(D\downarrow U)$.
The <general adjoint functor theorem> now supplies a <left adjoint> to $U$. Conversely, a <right adjoint> preserves small <categorical limits>, either by the <adjunction> <hom-set> bijections and the <hom-set detection of categorical limits>, or directly from their <universal properties>. Hence \b[the stated special theorem is proved in both directions].
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