---
title: Right Adjoints Preserve Limits (RAPL)
module: Adjoints, Representables, and Limits Together
moduleNumber: 6
lessonNumber: 3
order: 603
summary: >
  A functor with a left adjoint preserves every limit that exists, and dually a
  functor with a right adjoint preserves colimits. The proof is a four-line chain
  of natural isomorphisms through the adjunction and the continuity of
  representables. The theorem yields product-and-exponential arithmetic in Set,
  another proof that limits commute with limits, and a standard test for proving
  that a functor has no adjoint.
topics: [Adjoints, Representables, and Limits Together]
sources:
  - book: Leinster
    ref: "§6.3 Interactions between adjoint functors and limits; Theorem 6.3.1 and Examples 6.3.2–6.3.5"
  - book: Simmons
    ref: "Ch. 5 — Adjunctions (supporting examples of free-forgetful pairs)"
draft: false
---

A set-valued functor with a left adjoint is representable, and
[representables preserve limits](/category-theory/adjoints-limits/presheaf-limits-colimits),
so any set-valued functor with a left adjoint preserves limits. The restriction
to set-valued functors is unnecessary. The general statement, abbreviated
**RAPL**, is used constantly, both positively (a right adjoint automatically
preserves limits) and negatively (a functor that fails to preserve some limit
cannot have a left adjoint).[^leinster-631]

## The theorem

> **Theorem (RAPL).** Let $F \dashv G$ be an
> [adjunction](/category-theory/adjunctions/adjunctions) with
> $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{A}$. Then
> $F$ preserves colimits and $G$ preserves limits.

> **Proof.** By duality it is enough to prove the statement for $G$. Let
> $D : \mathcal{I} \to \mathcal{B}$ be a diagram with a limit. Then, naturally in
> $A \in \mathcal{A}$,
> $$
> \begin{aligned}
> \mathcal{A}\bigl(A, \, G(\textstyle\lim D)\bigr)
> &\;\cong\; \mathcal{B}\bigl(F(A), \, \textstyle\lim D\bigr)
>   && \text{adjointness} \\[2pt]
> &\;\cong\; \lim \, \mathcal{B}\bigl(F(A), \, D\bigr)
>   && \text{representables preserve limits} \\[2pt]
> &\;\cong\; \lim \, \mathcal{A}\bigl(A, \, G \circ D\bigr)
>   && \text{adjointness again} \\[2pt]
> &\;\cong\; \mathrm{Cone}(A, \, G \circ D)
>   && \text{cones as a limit of hom-sets.}
> \end{aligned}
> $$
> So $G(\lim D)$ represents the functor $\mathrm{Cone}(-, G \circ D)$, and a
> [representation of the cone functor is a limit](/category-theory/adjoints-limits/limits-via-adjoints):
> $G(\lim D)$ is a limit of $G \circ D$, with the image of the original limit cone
> as its limit cone. Each of the four isomorphisms is a result from an earlier
> lesson. $\blacksquare$

$$
% caption: RAPL: a right adjoint $G$ carries a limit cone in $\mathcal{B}$ to a
% limit cone in $\mathcal{A}$; the mediating map for any competing cone is found
% by transposing across the adjunction.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% B side
\node (L)  at (1.2,1.7)  {$\lim D$};
\node (D1) at (0,0)      {$D(I)$};
\node (D2) at (2.4,0)    {$D(J)$};
\draw[->, black] (L) -- (D1);
\draw[->, black] (L) -- (D2);
\draw[->, black] (D1) -- (D2);
\node[black, anchor=north] at (1.2,-0.5) {limit cone in $\mathcal{B}$};
% arrow between
\draw[->, acc, very thick] (3.6,0.85) -- (5.0,0.85) node[midway, above, black]{$G$};
% A side
\begin{scope}[xshift=6.4cm]
  \node[acc] (GL)  at (1.2,1.7) {$G(\lim D)$};
  \node (G1) at (0,0)   {$G D(I)$};
  \node (G2) at (2.4,0) {$G D(J)$};
  \draw[->, acc, thick] (GL) -- (G1);
  \draw[->, acc, thick] (GL) -- (G2);
  \draw[->, black] (G1) -- (G2);
  \node[black, anchor=north] at (1.2,-0.5) {still a limit cone};
\end{scope}
\end{tikzpicture}
$$

The dual statement has its own name: **left adjoints preserve colimits** (LAPC).
A left adjoint is characterized by maps _out_ of its values, and colimits are
defined by maps out; the symmetry is exact.

### The comparison map

"Preserves limits" has a precise canonical form. Suppose both $\lim D$ in
$\mathcal{B}$ and $\lim (G \circ D)$ in $\mathcal{A}$ exist. Applying $G$ to the
limit cone of $D$ gives a cone on $G \circ D$ with vertex $G(\lim D)$, which
factors uniquely through the limit cone of $G \circ D$:
$$
\kappa \;:\; G\bigl(\textstyle\lim D\bigr) \longrightarrow \lim \, (G \circ D),
\qquad
q_I \circ \kappa = G(p_I) \text{ for all } I,
$$
where $p_I$ and $q_I$ are the two families of projections. The functor $G$
preserves the limit of $D$ exactly when this **comparison map** $\kappa$ is an
isomorphism. RAPL asserts that for a right adjoint, $\kappa$ is invertible for
every diagram whose limit exists; the four-line proof above produces the inverse
by transposing cones across the adjunction. The comparison map is also what a
failed preservation looks like in practice: for the forgetful
$U : \mathbf{Grp} \to \mathbf{Set}$ and a coproduct diagram, the corresponding
co-comparison map $U(\mathbb{Z}) + U(\mathbb{Z}) \to U(\mathbb{Z} \ast
\mathbb{Z})$ identifies the two identity elements and hits only the one-syllable
words of the free product, missing every genuinely mixed word, nowhere near a
bijection.

$$
% caption: The comparison map (written $\kappa$ in the prose): $G$ applied to the
% limit cone of $D$ factors through the limit of $G D$, and $G$ preserves the
% limit exactly when this map is invertible.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\node[acc] (GL) at (0,1.6) {$G(\lim D)$};
\node (LG)  at (0,-0.2)    {$\lim (G D)$};
\node (GDI) at (3.8,0.7)   {$G D(I)$};
\draw[->, acc, thick] (GL) -- (LG) node[midway, left]{compare};
\draw[->, black] (GL) -- (GDI) node[midway, above right]{$G(p_I)$};
\draw[->, black] (LG) -- (GDI) node[midway, below right]{$q_I$};
\end{tikzpicture}
$$

| Functor | Preserves | Reason |
| --- | --- | --- |
| $G$ with a left adjoint | all limits that exist | RAPL |
| $F$ with a right adjoint | all colimits that exist | LAPC (dual) |
| $\mathcal{A}(A, -)$ | limits | representables preserve limits |
| $\mathcal{A}(-, A)$ | turns colimits into limits | contravariant dual |

## Consequences

### Forgetful functors and algebra

Forgetful functors from categories of algebras ($\mathbf{Grp}$,
$\mathbf{Vect}_k$, $\mathbf{Mon}$, $\mathbf{Ring}$) to $\mathbf{Set}$ have
[left adjoints](/category-theory/adjunctions/free-forgetful-adjunctions) — the
free constructions — so RAPL says they preserve all limits. This matches the
computations of the
[limits-in-concrete-categories lesson](/category-theory/limits-colimits/computing-limits):
the underlying set of a product of groups is the product of the underlying sets,
the underlying set of an equalizer is the equalizer of the underlying functions.
RAPL explains _why_ the underlying-set method works.

> **Worked example ($U : \mathbf{Grp} \to \mathbf{Set}$ preserves a product).**
> The forgetful functor $U$ has the free-group functor as a left adjoint, so it
> preserves limits. In $\mathbf{Grp}$ the product $\mathbb{Z}/2 \times
> \mathbb{Z}/3$ is a group of order $6$ (cyclic, $\cong \mathbb{Z}/6$), with
> componentwise multiplication and the two projections to the factors. Applying
> $U$,
> $$
> U(\mathbb{Z}/2 \times \mathbb{Z}/3) = \{0,1\} \times \{0,1,2\},
> \qquad
> |U(\mathbb{Z}/2 \times \mathbb{Z}/3)| = 6 = 2 \cdot 3 = |U(\mathbb{Z}/2)| \cdot |U(\mathbb{Z}/3)|,
> $$
> and the projections become the two set projections. The underlying set of the
> product is the product of the underlying sets. The same holds for equalizers:
> the equalizer of two homomorphisms $f, g : G \to H$ is the subgroup
> $\{\, x \in G : f(x) = g(x) \,\}$, whose underlying set is the equalizer of the
> underlying functions.

The same functors rarely have right adjoints, and correspondingly they rarely
preserve colimits. The underlying set of a coproduct of groups (the free
product) is far larger than the disjoint union of the underlying sets.[^leinster-632]

> **Worked example (A left adjoint that destroys a limit).** The free-monoid
> functor $F : \mathbf{Set} \to \mathbf{Mon}$ is a left adjoint, so it preserves
> colimits but need not preserve limits. The terminal object of $\mathbf{Set}$ is
> the one-element set $1$, an empty product; the terminal monoid is the trivial
> monoid $\{e\}$. But $F(1) = \mathbb{N}$, the free monoid on one generator under
> addition, which is infinite. So
> $$
> F(1) = \mathbb{N} \;\not\cong\; \{e\},
> $$
> and $F$ fails to send the terminal set to the terminal monoid. A left adjoint
> may destroy limits, exactly as RAPL allows.

### Arithmetic in Set

Every set $B$ gives an adjunction $(- \times B) \dashv (-)^B$ on $\mathbf{Set}$:
functions $A \times B \to C$ correspond to functions $A \to C^B$. RAPL and LAPC
applied to this pair produce the exponent laws of arithmetic.[^leinster-633]

- $- \times B$ is a left adjoint, so it preserves colimits, in particular finite
  sums:
  $$
  \varnothing \times B \cong \varnothing,
  \qquad
  (A_1 + A_2) \times B \;\cong\; (A_1 \times B) + (A_2 \times B).
  $$
- $(-)^B$ is a right adjoint, so it preserves limits, in particular finite
  products:
  $$
  1^B \cong 1,
  \qquad
  (A_1 \times A_2)^B \;\cong\; A_1^B \times A_2^B.
  $$

Taking cardinalities of finite sets recovers $0 \cdot b = 0$, distributivity,
$1^b = 1$, and $(a_1 a_2)^b = a_1^b \, a_2^b$. The natural numbers are the
isomorphism classes of finite sets, so these arithmetic identities are instances
of RAPL and LAPC.

$$
% caption: Distributivity as a colimit preserved by $- \times B$: the rectangle
% over $A_1 + A_2$ splits into the piece over $A_1$ and the piece over $A_2$,
% giving $(A_1 + A_2) \times B \cong (A_1 \times B) + (A_2 \times B)$.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[fill=acc!10, draw=acc, thick] (0,0) rectangle (2,2);
\draw[fill=acc!10, draw=acc, thick] (2,0) rectangle (5,2);
\node at (1,1)   {$A_1 B$};
\node at (3.5,1) {$A_2 B$};
\node[black, anchor=north] at (1,-0.1)   {$A_1$};
\node[black, anchor=north] at (3.5,-0.1) {$A_2$};
\node[black, anchor=east] at (-0.15,1)   {$B$};
\end{tikzpicture}
$$

### Commutation of limits via RAPL

When $\mathcal{A}$ has all limits of shape $\mathcal{I}$, the
[adjunction $\Delta \dashv \lim$](/category-theory/adjoints-limits/limits-via-adjoints)
makes $\lim : [\mathcal{I}, \mathcal{A}] \to \mathcal{A}$ a right adjoint. RAPL
applied to it says $\lim$ preserves limits, which says that limits of shape
$\mathcal{I}$ commute with all limits. This re-proves the
[Fubini-style commutation result](/category-theory/adjoints-limits/presheaf-limits-colimits)
for categories with all limits of one shape, with no computation.[^leinster-634]

## RAPL as a non-existence test

The contrapositive is a standard test: if a functor fails to preserve even one
colimit, it has no right adjoint; if it fails to preserve one limit, it has no
left adjoint.

> **Proposition (No left adjoint for Field).** The forgetful functor
> $U : \mathbf{Field} \to \mathbf{Set}$ has no left adjoint.

Suppose $F : \mathbf{Set} \to \mathbf{Field}$ were left adjoint to $U$. Left
adjoints preserve colimits, and the empty set is the initial object of
$\mathbf{Set}$ (the empty colimit), so $F(\varnothing)$ would be an initial
field: a field with exactly one homomorphism to every field. No such field
exists. Field homomorphisms are injective and preserve $0 \ne 1$, so any map of
fields preserves characteristic; a single field cannot map to both
$\mathbb{Q}$ (characteristic $0$) and $\mathbb{F}_p$ (characteristic $p$).
Hence $\mathbf{Field}$ has no initial object and $F$ cannot exist. There is no
free field on a set.[^leinster-635]

$$
% caption: The non-existence argument: a left adjoint would send the initial set
% $\varnothing$ to an initial field $K = F(\varnothing)$, but characteristic obstructs
% any single field from mapping to all others ($\mathbf{Q}$ denotes the rationals,
% $\mathbf{F}_p$ the field with $p$ elements).
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\node (I)  at (0,0)     {$K$};
\node (Q)  at (3.2,0.9) {$\mathbf{Q}$};
\node (Fp) at (3.2,-0.9){$\mathbf{F}_p$};
\draw[->, acc, thick] (I) -- (Q) node[midway, above left]{char $0$};
\draw[->, acc, thick] (I) -- (Fp) node[midway, below left]{char $p$};
\node[black, anchor=west] at (4.1,0) {both required; impossible};
\end{tikzpicture}
$$

The same template settles many cases.

- **$U : \mathbf{Grp} \to \mathbf{Set}$ has no right adjoint.** A right adjoint
  would make $U$ preserve colimits, but $U$ sends the coproduct
  $\mathbb{Z} \ast \mathbb{Z}$ (free product) to a set much bigger than
  $U(\mathbb{Z}) + U(\mathbb{Z})$.
- **The Yoneda embedding has no left adjoint in general.** It
  [fails to preserve colimits](/category-theory/adjoints-limits/presheaf-limits-colimits)
  — the initial object goes to a non-initial presheaf — so it cannot be a right
  adjoint. (Preserving limits is consistent with being a right adjoint but does
  not prove it; the failure direction is the decisive one.)
- **A functor between posets with a lower adjoint preserves meets.** For
  order-preserving maps viewed as functors between
  [thin categories](/category-theory/foundations/examples-of-categories), limits
  are meets and RAPL reads: a map with a lower (left) adjoint preserves all meets
  that exist. This is the classical Galois-connection fact.

> **Worked example (Floor and ceiling around $\mathbb{Z} \hookrightarrow
> \mathbb{R}$).** Let $\iota : \mathbb{Z} \hookrightarrow \mathbb{R}$ be the
> inclusion, both sets ordered as usual. The ceiling is left adjoint to $\iota$
> and the floor is right adjoint:
> $$
> \lceil x \rceil \le n \iff x \le \iota(n),
> \qquad
> n \le \lfloor x \rfloor \iff \iota(n) \le x .
> $$
> By RAPL and LAPC the floor (a right adjoint) preserves every meet that exists
> and the ceiling (a left adjoint) preserves every join. Finite meets and joins
> are uninformative in a total order, where $\min(x, y)$ is one of $x$ and $y$,
> so any monotone map preserves them. The content appears for infinite families.
> The floor preserves the infimum of any bounded family,
> $$
> \lfloor \inf_n x_n \rfloor = \inf_n \lfloor x_n \rfloor,
> $$
> as RAPL requires. The ceiling does not: with $x_n = 1 + \tfrac{1}{n}$,
> $$
> \Bigl\lceil \, \inf_n \bigl(1 + \tfrac{1}{n}\bigr) \Bigr\rceil = \lceil 1 \rceil = 1,
> \qquad
> \inf_n \, \Bigl\lceil 1 + \tfrac{1}{n} \Bigr\rceil = \inf_n 2 = 2 ,
> $$
> so the ceiling fails to preserve an infinite meet, consistent with its being
> only a left adjoint. Dually the ceiling preserves suprema and the floor fails
> on them.

$$
% caption: The Galois connection ceiling $\dashv$ inclusion $\dashv$ floor
% between the integers and the reals; RAPL makes the floor preserve meets and
% LAPC makes the ceiling preserve joins.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\node (Z) at (0,0)   {$\mathbf{Z}$};
\node (R) at (4.4,0) {$\mathbf{R}$};
\draw[->, black, thick] (Z) -- (R) node[midway, above]{inclusion};
\draw[->, acc, thick] (R.north west) to[bend right=28] node[midway, above]{ceiling} (Z.north east);
\draw[->, acc, thick] (R.south west) to[bend left=28] node[midway, below]{f\/loor} (Z.south east);
\node[black] at (2.2,1.5) {left adjoint: preserves joins};
\node[black] at (2.2,-1.5) {right adjoint: preserves meets};
\end{tikzpicture}
$$

```algorithm
caption: $\textsc{RuleOutAdjoint}(G)$ — testing a functor for a possible left adjoint
find a limit shape $\mathcal{I}$ and a diagram $D$ whose limit exists in the source
compute $G(\lim D)$ and $\lim (G \circ D)$ in the target
if the canonical comparison map is not an isomorphism then
  report $G$ has no left adjoint // by RAPL
else
  report inconclusive // preservation is necessary, not sufficient
```

The inconclusive branch is real: preserving limits does not by itself produce a
left adjoint. The unique functor $\mathcal{B} \to \mathbf{1}$ preserves every
limit, yet it has a left adjoint exactly when $\mathcal{B}$ has an initial
object, which plenty of complete-looking categories lack. The extra hypotheses
under which preservation _does_ suffice are the subject of the
[adjoint functor theorems](/category-theory/adjoints-limits/adjoint-functor-theorem).

## Scope of the theorem

- **RAPL preserves; it does not create.** If $\mathcal{B}$ lacks the limit of
  $D$, RAPL says nothing about a limit of $G \circ D$ appearing in
  $\mathcal{A}$. Compare
  [creation of limits](/category-theory/limits-colimits/limits-and-functors),
  which is a stronger property enjoyed by, for example, forgetful functors of
  algebras.
- **The converse fails without hypotheses.** Limit preservation is necessary for
  being a right adjoint, not sufficient (the $\mathcal{B} \to \mathbf{1}$
  example).
- **Each adjoint preserves only its own side.** A right adjoint may destroy
  colimits and a left adjoint may destroy limits; free functors typically
  destroy products (the free monoid on a one-element set is infinite, while the
  free monoid functor applied to a terminal object would need to be terminal to
  preserve it).

| Property of $G : \mathcal{B} \to \mathcal{A}$ | Implies left adjoint? | Implied by left adjoint? |
| --- | --- | --- |
| Preserves all limits | no (needs AFT hypotheses) | yes (RAPL) |
| Preserves some limits | no | yes, those included |
| Fails to preserve one limit | rules it out | — |

The converse direction — when limit preservation on $\mathcal{B}$ and $G$
guarantees a left adjoint — is settled by the
[general adjoint functor theorem](/category-theory/adjoints-limits/adjoint-functor-theorem),
a size condition whose proof runs through the comma-category description of
[adjunctions via universal arrows](/category-theory/adjunctions/adjunctions-via-universal-arrows).

[^leinster-631]: **Leinster**, _Basic Category Theory_, §6.3, Theorem 6.3.1 — right adjoints preserve limits, proved by the chain of natural isomorphisms through adjointness, continuity of representables, and the cone lemma.
[^leinster-632]: **Leinster**, _Basic Category Theory_, §6.3, Example 6.3.2 — forgetful functors from categories of algebras have left adjoints and so preserve limits, but rarely have right adjoints or preserve colimits.
[^leinster-633]: **Leinster**, _Basic Category Theory_, §6.3, Example 6.3.3 — the adjunction $(- \times B) \dashv (-)^B$ on $\mathbf{Set}$ and the resulting isomorphisms mirroring the arithmetic of natural numbers.
[^leinster-634]: **Leinster**, _Basic Category Theory_, §6.3, Example 6.3.4 — applying RAPL to $\Delta \dashv \lim$ shows limits commute with limits, recovering Proposition 6.2.8 for categories with all limits of one shape.
[^leinster-635]: **Leinster**, _Basic Category Theory_, §6.3, Example 6.3.5 — the forgetful functor from fields to sets has no left adjoint because $\mathbf{Field}$ has no initial object, characteristic being the obstruction.