Freyd's adjoint functor theorem

Remarks

A right adjoint preserves limits, and the unit $\eta_c$ alone is a solution set, which gives one direction. For the other, fix $c$ and consider the comma category $c \downarrow G$. It is complete, because $\mathcal{D}$ is complete and $G$ preserves limits. The product of a solution set is a weakly initial object, and the joint equalizer of all its endomorphisms is an initial object $(F c, \eta_c)$. These initial objects define the left adjoint $F$.

Example: the categories of groups, rings, modules and other algebraic structures are complete, and their forgetful functors to sets preserve limits. A cardinality bound on the subalgebra generated by a set gives the solution set condition. So the theorem gives free algebras without writing them down.

The solution set condition cannot be dropped. The forgetful functor from complete Boolean algebras to sets preserves limits, but has no left adjoint.

Going deeper

For a student

The theorem is due to Peter Freyd and is often called the general adjoint functor theorem. In practice, completeness and preservation of limits are easy to check, and the work is in the solution set condition. The special adjoint functor theorem replaces the solution set condition by conditions on $\mathcal{D}$ alone, such as a cogenerating set. In locally presentable categories the theorem simplifies further: a functor between them is a right adjoint exactly when it preserves limits and is accessible.