---
tag: T6A7JK3J98
kind: statement
role: theorem
name: Right adjoints preserve limits
msc: [18A40, 18A35]
uses: [TZHF6ZM92S, CHS5AFJGWE, H4MY8JN8W4]
about: [TZHF6ZM92S, H4MY8JN8W4]
sources:
  - url: https://en.wikipedia.org/wiki/Adjoint_functors
---

## Statement

Let $G \colon \mathcal{D} \to \mathcal{C}$ be a functor that has a [[left adjoint|TZHF6ZM92S]] $F \colon \mathcal{C} \to \mathcal{D}$.
Then $G$ [[preserves|H4MY8JN8W4]] every [[limit|CHS5AFJGWE]] that exists in $\mathcal{D}$.
Dually, $F$ preserves every colimit that exists in $\mathcal{C}$.

## Remarks

Let $(L, \lambda)$ be a limit cone of a diagram $K \colon J \to \mathcal{D}$.
For an object $c$, cones from $c$ over $G K$ match, by the adjunction, with cones from $F c$ over $K$.
These match with morphisms $F c \to L$, and so with morphisms $c \to G L$.
So $(G L, G \lambda)$ is a limit cone.

Example: the forgetful functor from groups to sets has a left adjoint, the free group.
So the underlying set of a product or an equalizer of groups is the product or the equalizer of the underlying sets.
The free group functor preserves coproducts: the free group on $X \sqcup Y$ is the free product of the free groups on $X$ and on $Y$.

The theorem gives a test for non-existence.
The forgetful functor from groups to sets does not send the initial trivial group to the initial empty set, so it has no right adjoint.

## Explanations

### L1 human

If a functor has a left adjoint, it behaves well with products, kernels and all other limits.
For example, forgetting the group structure of a product of groups leaves the product of the sets.
Dually, free constructions turn sums into sums.

### L2 human

This is the most used property of adjoint functors.
It explains many familiar facts at once, such as that tensor products distribute over direct sums, since $- \otimes M$ is left adjoint to $\Hom(M, -)$.
It is also the necessary half of the adjoint functor theorems.
Those theorems show that, under size conditions, preserving limits is enough for a functor to have a left adjoint.