---
tag: E33H1J8S53
kind: statement
role: theorem
name: Special adjoint functor theorem
msc: [18A40, 18A35]
uses: [FWGPKXCPB7, XVG7FPG227, CK08JW32WB, H4MY8JN8W4, 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.
Assume that $\mathcal{D}$ is [[well-powered|XVG7FPG227]] and has a cogenerating set.
Let $\mathcal{C}$ be a category 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 $G$ [[preserves limits|H4MY8JN8W4]] of small diagrams.

## Remarks

Here a set $Q$ of objects is cogenerating when, for any two different morphisms $f, g \colon a \to b$, some $q \colon b \to x$ with $x \in Q$ has $q f \neq q g$.
This is the dual of a [[generator|CK08JW32WB]].
The proof constructs an initial object of each comma category $c \downarrow G$.
Let $P$ be the product of copies of $x$, one for each $x \in Q$ and each morphism $c \to G x$.
Then one intersects all subobjects $S$ of $P$ such that the induced map $c \to G P$ factors through $G S$.
Well-poweredness makes this intersection a small limit.

Example: the category of compact Hausdorff spaces is complete and well-powered, and the unit interval $[0, 1]$ is a cogenerator.
The inclusion into topological spaces preserves limits.
So it has a left adjoint, the [[Stone–Čech compactification|D9MWP5C85W]].

## Explanations

### L1 human

This theorem is another test for a left adjoint.
Instead of a size condition on the functor, it asks for size conditions on the category where the functor starts.
Then preserving limits is all that is needed.
Its most famous use is to show that every space has a Stone–Čech compactification.

### L2 human

Compared with Freyd's general theorem, the solution set condition is replaced by well-poweredness and a cogenerating set in $\mathcal{D}$.
These conditions concern only $\mathcal{D}$, so they are checked once for all functors out of $\mathcal{D}$.
The dual theorem gives right adjoints of colimit-preserving functors out of a cocomplete, co-well-powered category with a generating set.
For instance, every colimit-preserving functor from a category of presheaves on a small category has a right adjoint.