---
tag: FX88V4XW6F
kind: statement
role: theorem
name: Freyd's adjoint functor theorem
msc: [18A40, 18A35]
uses: [FWGPKXCPB7, H4MY8JN8W4, CKTDR0VE8D, TZHF6ZM92S]
about: [TZHF6ZM92S]
sources:
  - url: https://en.wikipedia.org/wiki/Adjoint_functors
---

## Statement

Let $\mathcal{D}$ be a [[complete category|FWGPKXCPB7]] whose hom-sets are sets, and let $G \colon \mathcal{D} \to \mathcal{C}$ be a functor.
Then $G$ has a [[left adjoint|TZHF6ZM92S]] if and only if both of these hold:

1. $G$ [[preserves limits|H4MY8JN8W4]] of small diagrams.
2. $G$ satisfies the [[solution set condition|CKTDR0VE8D]].

## 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.

## Explanations

### L1 human

The theorem gives a test for when a functor has a left adjoint.
The necessary condition, that the functor preserve limits, is almost sufficient.
One only needs an extra size condition, which says that a set of test objects is enough.
It is a general way to show that free objects and similar universal constructions exist.

### L2 human

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.