= Solution
The <Special adjoint functor theorem> says: if $\mathcal C$ is a <locally small category>, a <complete category>, a <well-powered category>, and has a <small cogenerating family>, then a <functor> $U:\mathcal C\to\mathcal D$ into any <locally small category> has a <left adjoint> if and only if it preserves small <categorical limits>.
A <small cogenerating family> $(Q_i)_{i\in I}$, with $I$ a <set>, distinguishes unequal parallel <morphisms> by postcomposition: for $f\ne g:A\to B$, some $q:B\to Q_i$ has $qf\ne qg$. Being <well-powered> means that the <subobjects> of each object form a <set> up to the usual equivalence of <monomorphisms>. \b[These hypotheses make the solution-set condition automatic.]
The dual formulation uses a <cocomplete category>, being a <well-copowered category> and having a <small generating family>, and concludes that every small-colimit-preserving <functor> has a <right adjoint>.
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