---
title: Limits as Adjoints and as Representables
module: Adjoints, Representables, and Limits Together
moduleNumber: 6
lessonNumber: 1
order: 601
summary: >
  A cone on a diagram is a natural transformation from a constant diagram, so a
  limit is a representation of the cone functor and, equivalently, a value of the
  right adjoint to the diagonal functor. We prove both rephrasings, derive
  uniqueness and functoriality of limits from them, and record the dual statement
  that a colimit is the left adjoint to the diagonal.
topics: [Adjoints, Representables, and Limits Together]
sources:
  - book: Leinster
    ref: "Ch. 6 — Adjoints, representables and limits; §6.1 Limits in terms of representables and adjoints"
draft: false
---

Three formalisms express the same universal property:
[adjunctions](/category-theory/adjunctions/adjunctions),
[representable functors](/category-theory/representables-yoneda/representable-functors),
and [limits](/category-theory/limits-colimits/limits). Anything sayable in one is
sayable in the others. Rewriting the definition of a limit as a representation,
and again as an adjoint, recovers uniqueness, functoriality, and the dual
statement for colimits from general facts about representations and
adjoints.[^leinster-open]

A cone is a natural transformation whose source is constant.

## The diagonal functor

Fix a small category $\mathcal{I}$ (the _shape_) and a category $\mathcal{A}$.
For each object $A \in \mathcal{A}$ there is a functor
$\Delta A : \mathcal{I} \to \mathcal{A}$ that is constant: it sends every object
of $\mathcal{I}$ to $A$ and every map to $1_A$. Sending $A$ to this constant
diagram, and a map $g : A \to A'$ to the natural transformation with every
component equal to $g$, defines a functor into the
[functor category](/category-theory/foundations/natural-transformations).

> **Definition (Diagonal functor).** For a small category $\mathcal{I}$ and a
> category $\mathcal{A}$, the **diagonal functor** is
> $$
> \Delta : \mathcal{A} \longrightarrow [\mathcal{I}, \mathcal{A}],
> $$
> sending each object $A$ to the constant diagram $\Delta A$ with value $A$ and
> each map $g$ to the natural transformation all of whose components are $g$.

The name comes from the simplest shape. When $\mathcal{I}$ is the discrete
category on two objects, $[\mathcal{I}, \mathcal{A}] = \mathcal{A} \times
\mathcal{A}$ and $\Delta(A) = (A, A)$, the diagonal of the product. For a general
shape, $\Delta A$ is the constant diagram at $A$.

$$
% caption: The diagonal functor $\Delta$ carries an object $A$ to the diagram
% that is constant at $A$, with every internal map the identity.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% left: object A in the category A
\node[draw, circle, minimum size=8mm] (A) at (0,0) {$A$};
\node[font=\scriptsize, black, anchor=south] at (0,0.7) {$\mathcal{A}$};
% arrow
\draw[->, acc, very thick] (1.1,0) -- (3.1,0) node[midway, above, black]{diagonal};
% right: constant diagram in [I,A]
\begin{scope}[xshift=4.3cm]
  \node[draw, circle, minimum size=8mm] (c1) at (0,0.9) {$A$};
  \node[draw, circle, minimum size=8mm] (c2) at (1.7,0.9) {$A$};
  \node[draw, circle, minimum size=8mm] (c3) at (0.85,-0.7) {$A$};
  \draw[->, black] (c1) -- (c2) node[midway, above]{$1_A$};
  \draw[->, black] (c1) -- (c3);
  \draw[->, black] (c2) -- (c3);
  \node[font=\scriptsize, black, anchor=south] at (0.85,1.55) {$[\mathcal{I}$, $\mathcal{A}]$};
\end{scope}
\end{tikzpicture}
$$

## Cones as natural transformations

A [cone](/category-theory/limits-colimits/limits) on a diagram $D : \mathcal{I}
\to \mathcal{A}$ with vertex $A$ is a family of maps $f_I : A \to D(I)$, one for
each object $I \in \mathcal{I}$, compatible with the maps of the diagram:
$D(u) \circ f_I = f_J$ for every $u : I \to J$ in $\mathcal{I}$. Compare this to
the naturality condition for a transformation $\Delta A \Rightarrow D$. A
component at $I$ is a map $(\Delta A)(I) = A \to D(I)$, and naturality at
$u : I \to J$ is exactly $D(u) \circ f_I = f_J \circ 1_A = f_J$. The two
conditions coincide.

> **Proposition (Cones are transformations).** For a diagram $D : \mathcal{I}
> \to \mathcal{A}$ and an object $A$, a cone on $D$ with vertex $A$ is the same
> thing as a natural transformation $\Delta A \Rightarrow D$. Writing
> $\mathrm{Cone}(A, D)$ for the set of such cones,
> $$
> \mathrm{Cone}(A, D) = [\mathcal{I}, \mathcal{A}](\Delta A, D).
> $$

$$
% caption: A cone with vertex $A$ (left) is precisely a natural transformation
% from the constant diagram $\Delta A$ to $D$ (right); the leg conditions are the
% naturality squares.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% left: cone
\node (A)  at (1,1.6) {$A$};
\node (DI) at (0,0)   {$D(I)$};
\node (DJ) at (2,0)   {$D(J)$};
\draw[->, acc, thick] (A) -- (DI) node[midway, left]{$f_I$};
\draw[->, acc, thick] (A) -- (DJ) node[midway, right]{$f_J$};
\draw[->, black] (DI) -- (DJ) node[midway, below]{$D(u)$};
\node[black, anchor=north] at (1,-0.6) {cone};
% right: transformation
\begin{scope}[xshift=6cm]
  \node (a1) at (0,1.4) {$A$};
  \node (a2) at (2,1.4) {$A$};
  \node (b1) at (0,0)   {$D(I)$};
  \node (b2) at (2,0)   {$D(J)$};
  \draw[->, black] (a1) -- (a2) node[midway, above]{$1_A$};
  \draw[->, acc, thick] (a1) -- (b1) node[midway, left]{$f_I$};
  \draw[->, acc, thick] (a2) -- (b2) node[midway, right]{$f_J$};
  \draw[->, black] (b1) -- (b2) node[midway, below]{$D(u)$};
  \node[black, anchor=north] at (1,-0.6) {naturality square};
\end{scope}
\end{tikzpicture}
$$

Because $\mathrm{Cone}(A, D) = [\mathcal{I}, \mathcal{A}](\Delta A, D)$, the set
of cones is functorial in both arguments: contravariantly in the vertex $A$
(precompose a cone with a map $A' \to A$) and covariantly in the diagram $D$
(postcompose along a map $D \to D'$). Fixing $D$ and letting the vertex vary
gives the functor of interest,
$$
\mathrm{Cone}(-, D) : \mathcal{A}^{\mathrm{op}} \longrightarrow \mathbf{Set}.
$$

## Limits as representations

A limit cone on $D$ is a cone through which every other cone factors uniquely.
That universal property is just the condition for a
[representation](/category-theory/representables-yoneda/representable-functors)
of the cone functor: a universal element of $\mathrm{Cone}(-, D)$.[^leinster-611]

> **Proposition (Limit as representation).** Let $\mathcal{I}$ be small,
> $\mathcal{A}$ a category, and $D : \mathcal{I} \to \mathcal{A}$ a diagram.
> Limit cones on $D$ correspond one-to-one with representations of
> $$
> \mathrm{Cone}(-, D) : \mathcal{A}^{\mathrm{op}} \longrightarrow \mathbf{Set},
> $$
> the representing object being the limit vertex. Briefly: a limit of $D$ is a
> representation of $[\mathcal{I}, \mathcal{A}](\Delta -, D)$.

> **Proof.** A representation of a $\mathbf{Set}$-valued functor is a universal
> element, and a universal element of $\mathrm{Cone}(-, D)$ is a cone through
> which every cone factors uniquely,[^leinster-611] which is the definition of a
> limit cone. $\blacksquare$

Unwinding the representation gives a natural isomorphism. If $D$ has a limit,
then
$$
\mathrm{Cone}(A, D) \;\cong\; \mathcal{A}\bigl(A, \textstyle\lim D\bigr)
\qquad \text{naturally in } A.
$$
Left to right, a cone $(f_I)_{I}$ maps to its unique mediating map $m : A \to
\lim D$ through the limit cone; right to left, a map $g : A \to \lim D$ maps to
the cone $(p_I \circ g)_I$, where $p_I : \lim D \to D(I)$ are the limit
projections. The two assignments are mutually inverse by the universal property.

$$
% caption: The representation of the cone functor: cones with vertex $A$
% correspond to maps $A \to \lim D$, recovered by composing with the projections.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\node (A)   at (0,0)     {$A$};
\node (L)   at (2.6,0)   {$\lim D$};
\node (DI)  at (5.2,0.9) {$D(I)$};
\node (DJ)  at (5.2,-0.9){$D(J)$};
\draw[->, acc, thick] (A) -- (L) node[midway, above]{$m$};
\draw[->, black] (L) -- (DI) node[midway, above left]{$p_I$};
\draw[->, black] (L) -- (DJ) node[midway, below left]{$p_J$};
\draw[->, acc] (A) to[bend left=32] node[midway, above]{$f_I$} (DI);
\draw[->, acc] (A) to[bend right=32] node[midway, below]{$f_J$} (DJ);
\end{tikzpicture}
$$

> **Worked example (Binary product as $\Delta \dashv \lim$).** Take $\mathcal{I}$
> discrete on two objects and $\mathcal{A} = \mathbf{Set}$. A diagram $D$ is a
> pair of sets $(X, Y)$, and a cone with vertex $A$ is a natural transformation
> $\Delta A \Rightarrow D$: a pair of functions $A \to X$ and $A \to Y$, with no
> compatibility condition, since the shape has no non-identity maps. So
> $$
> \mathrm{Cone}(A, D) = \mathbf{Set}(A, X) \times \mathbf{Set}(A, Y).
> $$
> The limit is the cartesian product $X \times Y$ with its two projections, and
> the representation isomorphism reads
> $$
> \mathbf{Set}(A, X \times Y) \;\cong\; \mathbf{Set}(A, X) \times \mathbf{Set}(A, Y).
> $$
> For $A = 1$, $X = \{0, 1\}$, $Y = \{a, b, c\}$: a cone with vertex $1$ is a pair
> consisting of a point of $X$ and a point of $Y$, and there are $2 \cdot 3 = 6$
> of them, matching $|X \times Y| = 6$. Here $\lim$ is the product functor
> $- \times -$, and $\Delta \dashv \lim$ is the defining adjunction of binary
> products.

Two corollaries drop out. First, a representing object is
[unique up to isomorphism](/category-theory/universal-properties/universal-properties),
so limits are too.

> **Corollary (Uniqueness of limits).** Any two limits of the same diagram are
> isomorphic, by a unique isomorphism commuting with the projections.

Second, a terminal object is the limit of the empty diagram and a
[product](/category-theory/universal-properties/products-and-coproducts) is the
limit over a discrete shape, so both inherit the representation description
without separate argument.

## Functoriality of the limit

The representation isomorphism suggests varying $D$ as well as $A$. A map of
diagrams $\alpha : D \to D'$ (a natural transformation, with components
$\alpha_I : D(I) \to D'(I)$) should induce a map on limits, and it does.[^leinster-613]

> **Lemma (Induced map on limits).** Let $\mathcal{I}$ be small and
> $\alpha : D \Rightarrow D'$ a natural transformation between diagrams of shape
> $\mathcal{I}$, with limit cones $(p_I)$ on $D$ and $(p'_I)$ on $D'$. Then:
>
> - there is a unique map $\lim \alpha : \lim D \to \lim D'$ with
>   $p'_I \circ \lim\alpha = \alpha_I \circ p_I$ for every $I$;
> - any commuting square of cones over $\alpha$ induces a commuting square on the
>   factorizations through the limits.

> **Proof.** The composites $\alpha_I \circ p_I : \lim D \to D'(I)$ form a cone
> on $D'$ with vertex $\lim D$, so they factor uniquely through the limit cone of
> $D'$; that factorization is the forced value of $\lim\alpha$. $\blacksquare$

$$
% caption: The induced map on limits is the unique arrow making every projection
% square commute; here the map of diagrams $\alpha : D \Rightarrow D'$ is written
% $t$, the second diagram $D'$ is written $E$, and its projections are $q_I$.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\node (L)  at (0,1.4)  {$\lim D$};
\node (DI) at (2.6,1.4){$D(I)$};
\node (L2) at (0,0)    {$\lim E$};
\node (DI2)at (2.6,0)  {$E(I)$};
\draw[->, black] (L) -- (DI) node[midway, above]{$p_I$};
\draw[->, black] (L2) -- (DI2) node[midway, below]{$q_I$};
\draw[->, acc, thick] (L) -- (L2) node[midway, left]{$\lim t$};
\draw[->, acc, thick] (DI) -- (DI2) node[midway, right]{$t_I$};
\end{tikzpicture}
$$

The construction respects identities and composition, so $\lim$ is a functor.

## Limits as a right adjoint

Suppose $\mathcal{A}$ has _all_ limits of shape $\mathcal{I}$. Choose a limit
cone for each diagram $D$ and write its vertex $\lim D$. By the lemma, each map
$\alpha : D \to D'$ induces $\lim\alpha$, and these choices assemble into a
functor
$$
\lim : [\mathcal{I}, \mathcal{A}] \longrightarrow \mathcal{A}.
$$
The representation isomorphism now reads as an adjunction. On the one hand,
$[\mathcal{I}, \mathcal{A}](\Delta A, D) = \mathrm{Cone}(A, D)$; on the other,
$\mathrm{Cone}(A, D) \cong \mathcal{A}(A, \lim D)$ naturally in $A$. Taking the
diagram argument into account (via the second part of the lemma) makes the
isomorphism natural in $D$ as well.[^leinster-614]

> **Proposition (Limit is right adjoint to diagonal).** If $\mathcal{A}$ has all
> limits of shape $\mathcal{I}$, then $\lim : [\mathcal{I}, \mathcal{A}] \to
> \mathcal{A}$ is right adjoint to the diagonal functor:
> $$
> [\mathcal{I}, \mathcal{A}](\Delta A, D) \;\cong\; \mathcal{A}\bigl(A, \textstyle\lim D\bigr)
> \qquad \text{naturally in } A \text{ and } D,
> $$
> that is, $\Delta \dashv \lim$.

The choice of limit cones is non-canonical, but by
[uniqueness of adjoints](/category-theory/adjunctions/units-and-counits) a
different choice changes $\lim$ only up to natural isomorphism.

Dualizing reverses every arrow. A cocone on $D$ is a natural transformation
$D \Rightarrow \Delta A$, a colimit is a representation of the cocone functor,
and when all colimits of shape $\mathcal{I}$ exist the assignment
$\operatorname{colim} : [\mathcal{I}, \mathcal{A}] \to \mathcal{A}$ is _left_
adjoint to the diagonal.

> **Proposition (Colimit is left adjoint to diagonal).** If $\mathcal{A}$ has all
> colimits of shape $\mathcal{I}$, then
> $$
> \mathcal{A}\bigl(\operatorname{colim} D, A\bigr) \;\cong\;
> [\mathcal{I}, \mathcal{A}](D, \Delta A)
> \qquad \text{naturally in } A \text{ and } D,
> $$
> that is, $\operatorname{colim} \dashv \Delta$.

The diagonal has $\operatorname{colim}$ as a left adjoint and $\lim$ as a right
adjoint: $\operatorname{colim} \dashv \Delta \dashv \lim$.

$$
% caption: When a category has all limits and colimits of shape $\mathcal{I}$,
% the diagonal functor $\Delta$ sits between two adjoints:
% $\operatorname{colim} \dashv \Delta \dashv \lim$.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\node (F) at (0,0)   {$[\mathcal{I}$, $\mathcal{A}]$};
\node (A) at (4.4,0) {$\mathcal{A}$};
\draw[->, acc, thick] (F.north east) to[bend left=28] node[midway, above]{lim} (A.north west);
\draw[->, black, thick] (A) -- (F) node[midway, above]{diagonal};
\draw[->, acc, thick] (F.south east) to[bend right=28] node[midway, below]{colim} (A.south west);
\node[black] at (2.2,1.5) {right adjoint};
\node[black] at (2.2,-1.5) {left adjoint};
\end{tikzpicture}
$$

## Completeness in adjoint form

A category is [complete](/category-theory/limits-colimits/limits) when it has all
small limits. The adjoint rephrasing turns completeness into a statement about
the diagonal functor.

| Formalism | A limit of $D$ is | Existence of all limits of shape $\mathcal{I}$ means |
| --- | --- | --- |
| Explicit | a universal cone on $D$ | every diagram has a universal cone |
| Representable | a representation of $\mathrm{Cone}(-, D)$ | $\mathrm{Cone}(-, D)$ is representable for every $D$ |
| Adjoint | the value $\lim D$ of the right adjoint to $\Delta$ | $\Delta : \mathcal{A} \to [\mathcal{I}, \mathcal{A}]$ has a right adjoint |

The third row reduces "has all limits of shape $\mathcal{I}$" to "the diagonal
functor has a right adjoint", which lets general theorems about adjoints act on
limits. Applying the fact that
[right adjoints preserve limits](/category-theory/adjoints-limits/adjoints-preserve-limits)
to $\lim$ itself shows that
[limits commute with limits](/category-theory/adjoints-limits/presheaf-limits-colimits),
and the question of when a limit-preserving functor is a right adjoint is
answered by the
[adjoint functor theorem](/category-theory/adjoints-limits/adjoint-functor-theorem).

> **Worked example (Fixed points and orbits).** When $\mathcal{I}$ is a group
> $G$ (one object, every map invertible) and $\mathcal{A} = \mathbf{Set}$, a
> diagram is a $G$-set and the constant diagram $\Delta A$ carries the trivial
> action. A cone $\Delta A \Rightarrow D$ is a $G$-equivariant map out of a
> trivial $G$-set, so its image lands in the fixed points; the limit $\lim D$ is
> the fixed-point set $D^G$, and dually $\operatorname{colim} D$ is the orbit set
> $D/G$.
> Take $G = \mathbb{Z}/2 = \{e, \sigma\}$ acting on $D = \{1, 2, 3, 4\}$ with
> $\sigma$ swapping $1 \leftrightarrow 2$ and fixing $3$ and $4$. Then
> $$
> \lim D = D^G = \{3, 4\}, \qquad \operatorname{colim} D = D/G = \bigl\{\{1, 2\},\, \{3\},\, \{4\}\bigr\}:
> $$
> two fixed points and three orbits. The pair
> $\operatorname{colim} \dashv \Delta \dashv \lim$ specializes to orbits and
> fixed points.

[^leinster-open]: **Leinster**, _Basic Category Theory_, Ch. 6 opening — the three formalisms (adjointness, representability, limits) as three coordinate systems for universal properties, with the chapter devoted to translating between them.
[^leinster-611]: **Leinster**, _Basic Category Theory_, §6.1, Proposition 6.1.1 — a limit of $D$ is a representation of $\mathrm{Cone}(-, D)$, proved by identifying a representation with a universal cone via the corollary on representations as universal elements.
[^leinster-613]: **Leinster**, _Basic Category Theory_, §6.1, Lemma 6.1.3 — a map of diagrams induces a unique map of limits commuting with the projections, and the induced square on factorizations commutes.
[^leinster-614]: **Leinster**, _Basic Category Theory_, §6.1, Proposition 6.1.4 — when all limits of shape $\mathcal{I}$ exist, $\lim$ is a functor and is right adjoint to $\Delta$; the dual gives $\operatorname{colim} \dashv \Delta$.