Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 4 / b / 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. 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.