[ct/2] Universal Arrows, Yoneda Lemma, Limits


Paris, Chennai

Universal arrow

Let $S : A \rightarrow B$ be a functor. For the objects $a \in A, b \in B$, the universal arrow from $b$ to $S$ is defined as the pair $\langle r, u \rangle$ with object $r \in A$ and morphism $u : b \rightarrow Sr$:

$$ \begin{xy} \xymatrix{ b \ar[r]^u \ar@{}[d]|{=} & Sr \ar@{.>}[d]|{\exists!} \\ b \ar[r]^f & Sa } \end{xy} $$

Said differently, every arrow $f$ factors uniquely through $u$.

Examples:

  1. $\textbf{Vec}_K$ is defined as the vector space over field $K$. The forgetful functor $U : \textbf{Vec}_K \rightarrow \textbf{Set}$ is a function sending a vector space to its underlying elements. For any set $X$, there exists a $\textbf{Vec}_X$ with $X$ as the basis. This fact is illustrated by the function $j : X \rightarrow U(V_X)$, and there also exists $k : X \rightarrow U(W)$ obtained by linear transformation. This makes $j$ a universal functor.
  2. Free category $C$ over graph $G$ is given by the universal functor $k : C \rightarrow UG$. Similarly, we have universal functors for a free group or free R-module on a given set of generators.
  3. For integral domain $D$ and field of quotients $Q(D)$, there exists a monomorphism $j : D \rightarrow Q(D)$. The universal (forgetful) functor $\langle Q(D), j \rangle$ maps $\textbf{Fld} \rightarrow \textbf{Dom}_m$, from the category of fields to the category of domains, provided we take the arrows of $\textbf{Dom}_m$ to be monomorphisms of integral domains.

A `universal element` of functor $H : D \rightarrow \textbf{Set}$ is defined to be the pair $\langle r, e \rangle$ consisting of $r \in D$, $e \in Hr$, such that $\langle d, x \rangle$ consisting of $Hr \in D$ and arrow $f : r \rightarrow d$ implies that $(Hf)e = x$.

Consider the equivalence relation $E$ on set $S$, and the projection $p : S \rightarrow S/E$ sending every element in the set to its E-equivalence class:

$$ \begin{xy} \xymatrix{ S \ar@{}[d]|{=}\ar[r]^p & S/E \ar[d]^g \\ S \ar[r]^f & X } \end{xy} $$

This states that $\langle S/E, p \rangle$ is a universal element under the functor $H : \textbf{Set} \rightarrow \textbf{Set}$.

$f : G \rightarrow G/N$, the morphism that sends each group to its quotient group, is another example of the universal element for the functor $H : \textbf{Grp} \rightarrow \textbf{Set}$.

Relationship between universal arrow and universal element: An element $e$ can be considered universal arrow $H : \textbf{*} \rightarrow Hr \in \textbf{Ens}$ from the one-point set $\textbf{*}$; then $\langle r, e \rangle$ from $\{\textbf{*}\}$ to $H$ is a universal element. If $G : A \rightarrow B$ is a functor, then the pair $\langle r, u : b \rightarrow Sr \rangle$ is a universal arrow if $\langle r, u \in C(b, Sr) \rangle$ is a universal element of the functor $H = C(b, S-)$.

Projections $p : a \times b \rightarrow a$ and $q : a \times b \rightarrow b$ give us another example of universal arrows. Given any other pair of arrows $f : c \rightarrow a$, $g : \rightarrow b$, and $h : c \rightarrow a \times b$, we get $ph = f$ making $\langle p, q \rangle$ a `universal pair`. To make it a universal arrow, consider the diagonal functor $\Delta C = C \times C$ so that the pair $\langle f, g \rangle : \Delta c \rightarrow \langle a, b \rangle$; then $\langle p, q \rangle$ is a universal arrow from $\Delta$ to $a \times b$.

Yoneda lemma

The concept of universality can be formulated in terms of [hom-sets](/category/1#hom-sets) as follows. Given functor $S: A \rightarrow B$, the pair $\langle r, u : b \rightarrow Sr \rangle$ for $r \in A$ is universal if and only if the function $g : r \rightarrow a$ and the composition $Sg \circ u : b \rightarrow Sa$ satisfies the bijection of hom-sets $A(r, a) \cong B(b, Sa)$. This is natural in $A$. If $A, B$ have small hom-sets, then the functor $B(b, S-)$ to $\textbf{Set}$ is isomorphic to the covariant hom-functor $A(r, -)$. Such isomorphisms are called `representations`.

Let $D$ be a category with small hom-sets. Then, `representation` of functor $K : D \rightarrow \textbf{Set}$, given $r \in D$, is the pair $\langle r, \psi \rangle$ such that: $$\psi : D(r, -) \cong K$$

$r$ is called `representing object`, and $K$ is termed `representable`, when such a representation exists. A representation is hence just a covariant hom-functor $D(r, -)$.

Let $D$ be a category with small hom-sets, $r \in D$, and $\textbf{*}$ denote the one-point set. If $\langle r, u \rangle: \textbf{*} \rightarrow Kr$ is a universal arrow from $\textbf{*}$ to $K : D \rightarrow \textbf{Set}$, then function $\psi$ which, for every object of $D$, sends the arrow $g : r \rightarrow d$ to $K(g)(u\textbf{*}) \in Kd$ is a representation of $K$.

A universal arrow from $b$ to $S : A \rightarrow B$ can be written as the natural isomorphism $A(r, a) \cong B(b, Sa)$, or equivalently as representation of the functor $B(b, S-) : A \rightarrow \textbf{Set}$, or equally well as the universal element of the same functor.

Informally speaking, the `Yoneda lemma` states that an object can be recovered by knowing the maps into it. Formally, given category $D$ with small hom-sets, $r \in D$, functor $K : D \rightarrow \textbf{Set}$, there is a bijection: $$\text{Nat}(D(r, -), K) \cong Kr$$

which sends each NT $D(r, -) \overset{\bullet}{\rightarrow} K$ to $\alpha_r 1_r$, the identity $r \rightarrow r$. The proof is indicated as:

$$ \begin{xy} \xymatrix{ D(r, r)\ar[r]^{\alpha_r}\ar[d]_{K(d, f)} & K(r)\ar[d]^{K(f)} & r \ar[d]^f\\ D(r, d)\ar[r]^{\alpha_d} & K(d) & d } \end{xy} $$

The bijection $y : \text{Nat}(D(r, -), K) \cong Kr$ is the natural isomorphism $N \overset{\bullet}{\rightarrow} E : \textbf{Set}^D \times D \rightarrow \textbf{Set}$. The object function $r \mapsto D(r, -)$ and the arrow function: $$(f : s \rightarrow r) \mapsto D(f, -) : D(r, -) \mapsto D(s, -)$$

define a fully faithful functor $Y : D^{op} \rightarrow \textbf{Set}^{D}$, and this is called the `Yoneda functor`.

Coproducts and colimits

For any category $C$, the diagonal functor $\Delta : C \rightarrow C \times C$ is given by the object mapping $\Delta : c \rightarrow \langle c, c \rangle$ and morphism mapping $\Delta : f \rightarrow \langle f, f \rangle$. A universal arrow from object $\langle a, b \rangle$ to $\Delta$, given by $\langle a, b \rangle \rightarrow \langle c, c \rangle$ is called the `coproduct diagram`. When such a diagram exists, object $c$ is written as $a \sqcup b$:

$$ \begin{xy} \xymatrix{ a\ar[r]^i\ar[dr]_f & a \sqcup b \ar@{.>}[d]|{\exists! h} & b\ar[l]_j\ar[dl]^g \\ & d & } \end{xy} $$

The assignment $\langle f, g \rangle \mapsto h$ is a bijection that is natural in $d$: $$C(a, d) \times C(b, d) \cong C(a \sqcup b, d)$$

Examples include the disjoint union in $\textbf{Set}$, wedge product in $\textbf{Top}_*$, and the tensor product in $\textbf{CRng}$.

For `infinite coproducts`, replace $C^2$ with $C^X$, for any set $X$: $$C(\sqcup_x a_x, c) \cong \prod_x C(a_x, c)$$

If $a_x = b$ for all x, `copower` $\sqcup_X b$ is written as $X \bullet b$: $$C(X \bullet b, c) \cong C(b, c)^X$$

If $C$ has a null object such that $0: b \rightarrow z \rightarrow c$ is the zero arrow, `cokernel` of $f : a \rightarrow b$ is given by $u : b \rightarrow c$:

$$ \begin{xy} \xymatrix{ a\ar[r]^f & b \ar[r]^u\ar[dr]_h & c\ar@{.>}[d]|{\exists! h'} & uf = 0 \\ & & d & hf = 0 } \end{xy} $$

Given a pair of arrows $f, g : a \rightarrow b$, `coequalizer` $u$ can be defined as:

$$ \begin{xy} \xymatrix{ a\ar@<-.5ex>[r]_f\ar@<.5ex>[r]^g & b \ar[r]^u\ar[dr]_h & c\ar@{.>}[d]|{\exists! h'} & uf = ug \\ & & d & hf = hg } \end{xy} $$

Let $\downarrow \downarrow$ represent a category $C$ with two objects and two non-identity arrows between them. Then, functor category $C^{\downarrow \downarrow}$ can be formed:

$$ \begin{xy} \xymatrix{ a\ar@<-.5ex>[r]_f\ar@<.5ex>[r]^g\ar[d]_h & b\ar[d]^k \\ a'\ar@<-.5ex>[r]_{f'}\ar@<.5ex>[r]^{g'} & b' } \end{xy} $$

Having defined the diagonal functor $\Delta : C \rightarrow C^{\downarrow \downarrow}$, and object $c \in C$, consider:

$$ \begin{xy} \xymatrix{ a\ar@<-.5ex>[r]_f\ar@<.5ex>[r]^g\ar[d]_{kf} & b\ar[d]^k \\ c\ar@<-.5ex>[r]_1\ar@<.5ex>[r]^1 & c } \end{xy} $$

In other words, $k$ coequalizes $f, g$. A coequalizer of pair $\langle f, g \rangle$ is simply a universal arrow $\langle f, g \rangle \rightarrow \Delta$.

As an example, coequalizer in $\textbf{Set}$ for the set of functions $f, g : X \rightarrow Y$ is simply the projection $Y \rightarrow Y/E$ where $E$ is the least equivalence relation $E \subset Y \times Y$.

Given $f : a \rightarrow b, g : a \rightarrow c$, `pushout` is defined as the pair $\langle f, g \rangle$ such that:

$$ \begin{xy} \xymatrix{ \langle f, g \rangle \ar[d] & b\ar[d]^h & a\ar[d]^{hf = kg}\ar[l]_f\ar[r]^g & c\ar[d]^k \\ \Delta(s)& s & s\ar[l]_1\ar[r]^1 & s } \end{xy} $$

The `coproduct over $a$`, also called `cocartesian square` or `fibered sum` is written as: $$r = b \sqcup_a c = b \sqcup_{\langle f, g \rangle} c$$

The pushout of $\langle f, g \rangle$ always exists in $\textbf{Set}$; it is the disjoint union $a \sqcup b$ with elements identified with $fx$ and $gx$, for $x \in a$.

For $f: a \rightarrow b \in C$, the `cokernel pair` is defined as the pair of pushout of $f$ along with $f$. Indeed, there is some $r$ so that $u, v : b \rightarrow r$ are parallel arrows:

$$ \begin{xy} \xymatrix{ a\ar[r] & b\ar@<-.5ex>[r]_u\ar@<.5ex>[r]^v\ar@<-.5ex>[dr]_f\ar@<.5ex>[dr]^g & r\ar@{.>}[d]|{\exists! t} \\ & & s } \end{xy} $$

Let $C$ be a category, and $J$ be an index category. The diagonal functor $$\Delta : C \rightarrow C^J$$

sends each object $c$ to the constant functor $\Delta c$ - the functor which has the same value $c$ for each object $i \in J$, and the value $1_c$ at each arrow of $J$. Given $f : c \rightarrow c'$, $\Delta f : \Delta c \overset{\bullet}{\rightarrow} \Delta c'$ is an NT. Arrows $F : J \rightarrow C$ correspond to objects of $C^J$. The universal arrow $\langle r, u \rangle$ from $F$ to $\Delta$ is called a `colimit` (or `direct limit`, `inductive limit`) diagram for $F$; it consits of an object $r \in C$, usually written $r = \underset{\longrightarrow}{\text{Lim }} F$, or $r = \text{Colim } F$, along with an NT $u : F \overset{\bullet}{\rightarrow} \Delta r$ which is universal among NTs $\tau : F \overset{\bullet}{\rightarrow} \Delta c$. Pictorially, the following diagram commutes:

$$ \begin{xy} \xymatrix{ F_1\ar[d]^{\tau_1}\ar[r] & F_2\ar[d]^{\tau_2}\ar[r] & F_3\ar[d]^{\tau_3}\ar@<-.5ex>[r]\ar@<.5ex>[r] & F_4\ar[d]^{\tau_4} & F_5\ar[d]^{\tau_5}\ar[l] \\ c\ar@{}[r]|{=} & c\ar@{}[r]|{=} & c\ar@{}[r]|{=} & c\ar@{}[r]|{=} & c } \end{xy} $$

$\tau : F \overset{\bullet}{\rightarrow} \Delta c$ is often written as $\tau : F \overset{\bullet}{\rightarrow} c$, and is called the `cone` of $F$ to the base $c$:

$$ \begin{xy} \xymatrix{ F_i\ar[r]^{F_u}\ar[dr]_{\tau_i} & F_j\ar[d]^{\tau_j}\ar[r]^{F_v} & F_k\ar[dl]^{\tau_k} \\ & c & } \end{xy} $$

Alternatively, colimit of $F : J \rightarrow C$ consists of object $\underset{\longrightarrow}{\text{Lim }} F \in C$ along with the cone $\mu : F \overset{\bullet}{\rightarrow} \Delta (\underset{\longrightarrow}{\text{Lim }} F)$, from base $F$ to the vertex $\underset{\longrightarrow}{\text{Lim }} F$, which is a `universal cone` (or `limiting cone`).

As an example, consider $J = \omega = \{0 \rightarrow 1 \rightarrow 2 \rightarrow \ldots\}$, and functor $F : \omega \rightarrow \textbf{Set}$ which maps every arrow in $\omega$ to an inclusion map (subset in set). This functor is simply the inclusion $F_0 \subset F_1 \subset F_2 \subset \ldots$. The union $U$ of all sets, with cone given by inclusion map $F_n \rightarrow U$, is $\underset{\longrightarrow}{\text{Lim }} F$. For $J$ small, any $J \rightarrow \textbf{Set}$ has a colimit.

Limits and products

The notion of a limit is dual to that of a colimit. Let $C$ be a category, $J$ be an index set, and $\Delta : C \rightarrow C^J$ be a diagonal functor. Limit of functor $F : J \rightarrow C$ is defined as the universal arrow $\langle r, \nu \rangle$ from $\Delta$ to $F$; it consists of object $r \in C$, $r = \underset{\longleftarrow}{\text{Lim }} F = \text{Lim } F$, called the `limit object` (or `projective limit`, `inverse limit`) of $F$, and the NT $\nu : \Delta r \overset{\bullet}{\rightarrow} F$, which is universal among $\tau : \Delta c \overset{\bullet}{\rightarrow} F$ for $c \in C$. $\tau : c \overset{\bullet}{\rightarrow} F$ is then called cone to base $F$ from vertex $c$, pictured as:

$$ \begin{xy} \xymatrix{ c\ar[d]_{\tau_i}\ar[dr]|{\hole}_{\tau_j}\ar[r]^t & \underset{\longleftarrow}{\text{Lim }} F = \text{Lim } F\ar[d]^{\nu_j}\ar[dl]_{\nu_i} \\ F_i\ar[r]_{Fu} & F_j } \end{xy} $$

The properties of limits and colimits may be pictured as:

$$ \begin{xy} \xymatrix{ \underset{\longleftarrow}{\text{Lim }} F\ar[r]^\nu & F\ar@{}[d]|{=}\ar[r]^\mu & \underset{\longrightarrow}{\text{Lim }} F\ar@{.>}[d] \\ c\ar[r]^\tau\ar@{.>}[u] & F\ar[r]^\sigma & c } \end{xy} $$

For a discrete category $J = \{1, 2\}$ and category $C$, limit of functor $F : \{1, 2\} \rightarrow C$ consisting of pairs $\langle a, b \rangle$, is called `product` of $a, b$ and written as $a \times b$. Then, we have projections $p, q$ defined as: $$a \xleftarrow{p} a \times b \xrightarrow{q} b$$

They form a cone with vertex $a \times b$, and we have the bijection of sets natural in $c$: $$C(c, a \times b) \cong C(c, a) \times C(c, b)$$

We then define $h = (f, g) : c \rightarrow a \times b$, and $f, g$ are called the `components` of $h$. In $\textbf{Cat}, \textbf{Grp}, \textbf{Top}$, this corresponds to direct product.

`Infinite products`. When $J$ is a set (= discrete category, category with all arrows identities), $F : J \rightarrow C$ is simply a $J$-indexed family of objects $a_i \in C$, while cone with vertex $c$ and base $a_j$ is a $J$-indexed family of arrows $f_j : c \rightarrow a_j$. We have $f : c \rightarrow \prod_j a_j$ and the following bijection of sets natural in $c$: $$\prod_j C(c, a_j) \cong C(c, \prod_j a_j)$$

Products over any small set exist in $\textbf{Set}$, $\textbf{Grp}$, and $\textbf{Top}$; they are simply cartesian products.

If factors in a product are all equal, $a_j = b$ for all $j$, then $\prod_j a_j = \prod_j b$ is called `power`, and is written as $\prod_j b = b^J$. The following bijection of sets is natural in $c$: $$C(c, b)^J \cong C(c, b^J)$$

Given $J = \downarrow \downarrow$ and functor $F : \downarrow \downarrow \rightarrow C$ defined by parallel arrows $f, g : a \rightarrow b$, the limit point $d \in F$, when it exists, is called an `equalizer` or `difference kernel` of $f$ and $g$:

$$ \begin{xy} \xymatrix{ d\ar[r]^e & a\ar@<-.5ex>[r]_g\ar@<.5ex>[r]^f & b } \end{xy} $$

The limit arrow $e$ amounts to cone $a \leftarrow d \rightarrow b$ from vertex $d$. In $\textbf{Set}$, equalizer always exists; $d$ is the set $d \in b \mid fx = gx$, and $e : d \rightarrow b$ is an injection of this subset of $b$ into $b$. In $\textbf{Ab}$, the equalizer $d$ is kernel of difference homomorphism $f - g : a \rightarrow b$. Any equalizer $e$ is necessarily a monic.

For $J : \rightarrow \bullet \leftarrow$, functor $F : (\rightarrow \bullet \leftarrow) \rightarrow C$ is a pair of arrows $b \xrightarrow{g} a \xleftarrow{f} d$. The cone of such a functor is a pair of arrows from vertex $c$ in the following diagram:

$$ \begin{xy} \xymatrix{ c\ar@/_/[ddr]\ar@{.>}[dr]|{\exists!}\ar@/^/[drr] \\ & b \times_a d\ar[d]^p\ar[r]_q & d\ar[d]_f \\ & b\ar[r]^g & a } \end{xy} $$

The diagram also illustrates the universal cone formed by $c \rightarrow b \times_a d$, and the square is called `pullback square`. The product $b \times_a d$ is then called pullback or `fibered product`.

In $\textbf{Top}$, if $g : d \rightarrow a$ is a "fiber map" with base $a$, and $f$ is a continuous map into the base, then projection $p$ of the pullback is the `induced fiber map`. When it exists, the pullback of a pair of equal arrows $b \rightarrow a \leftarrow b$ is called the `kernel pair` of $f$. A limit of the empty functor in $C$ is a terminal object of $C$.

Categories with finite products

A category is said to have finite products if, for any finite number of objects $c_1, c_2, \ldots c_n$, there exists a product object $c_1 \times c_2 \times \ldots c_n$ and projections $p_i : c_1 \times c_2 \times \ldots c_n \rightarrow c_i$ for $i = 1 \ldots n$, with the usual universal property.

If $C$ has a terminal object $t$ and products $a \leftarrow a \times b \rightarrow b$ for any two objects $a, b$, then $C$ has all finite products. The products provide, by $\langle a, b \rangle \mapsto a \times b$, the bifunctor $C \times C \rightarrow C$. For any three objects, we have the isomorphism natural in $a, b, c$: $$\alpha = \alpha_{a, b, c} : (a \times b) \times c \cong a \times (b \times c)$$

For any object $a$, there are isomorphisms natural in $c$: $$\rho_a = \rho : t \times a \cong a \quad \lambda_a = \lambda : a \times t \cong a$$

The proof can be expressed as a commutative diagram:

$$ \begin{xy} \xymatrix{ & & b \\ a & a \times (b \times c)\ar[l]\ar[r]\ar[ur]\ar[dr] & b \times c\ar[u]\ar[d] \\ & & c } \end{xy} $$

The dual result holds for a coproduct. A coproduct diagram consists of $m$ injections $i_j : a_j \rightarrow a_1 \sqcup a_2 \sqcup \ldots \sqcup a_m$ and a map $f: a_1 \sqcup a_2 \sqcup \ldots \sqcup a_n \rightarrow c$ determined by its $m$ cocomponents $f \circ i_j : f_j = a_j \rightarrow c$ for $j = 1, 2, \ldots m$. If $C$ has both finite products and finite coproducts, the arrows $$a_1 \sqcup \ldots \sqcup a_m \rightarrow b_1 \times \ldots \times b_n$$

from coproducts to products is determined uniquely by an $m \times n$ matrix of arrows $f_{jk} = p_k f i_j : a_j \rightarrow b_k$. In any category with a null object $z$ and the zero arrow $a \rightarrow b$ through $z$, finite coproducts, and finite products, there is a canonical arrow of the coproduct to the product: $$a_1 \sqcup \ldots \sqcup a_n \rightarrow a_1 \times \ldots \times a_n$$

This arrow is precisely the identity matrix of order $n$. It may be an isomorphism (in $\textbf{Ab}, \textbf{R-Mod}$), a proper monic (in $\textbf{Top}, \textbf{Set}$), or a proper epi (in $\textbf{Grp}$).

Groups in categories

Let $C$ be a category with finite products and terminal object $c$. Then, a monoid is defined by the triple $\langle c, \mu : c \times c \rightarrow c, \eta : t \rightarrow c \rangle$ such that the following diagrams commute:

$$ \begin{xy} \xymatrix{ c \times (c \times c)\ar[r]^\alpha\ar[d]^{1 \times \mu} & (c \times c) \times c\ar[r]^{\mu \times 1} & c \times c\ar[d]^\mu \\ c \times c\ar[rr]^{\mu} & & c } \end{xy} $$

$$ \begin{xy} \xymatrix{ t \times c\ar[r]^{\eta \times 1}\ar[d]^\lambda & c \times c\ar[d]^\mu & c \times t\ar[l]_{1 \times \eta}\ar[d]^\rho \\ c\ar@{}[r]|{=} & c\ar@{}[r]|{=} & c } \end{xy} $$

where $\alpha$ is the associativity morphism. A group is then defined by the triple $\langle c, \mu, \eta \rangle$ together with the operation $\zeta : c \rightarrow c$ such that the following diagram commutes (with $\delta_c$ the diagonal):

$$ \begin{xy} \xymatrix{ c\ar[r]^{\delta_c}\ar[d] & c \times c\ar[r]^{1 \times \zeta} & c \times c\ar[d]^\mu \\ t\ar[rr]^\eta & & c } \end{xy} $$

This suggests that $\zeta$ sends each element $c$ to its right inverse. One can draw similar diagrams for any algebraic system.

If $C$ is a category with finite products, then object $c$ is a group iff the hom-functor $C(-, c)$ is a group in the functor category $\textbf{Set}^{C^{op}}$. Each multiplication $\mu$ of $C$ determins a corresponding multiplication $\bar{\mu}$ in the hom-set $C(-, c) : C^{op} \rightarrow \textbf{Set}$ as the composite

$$ C(-, c) \times C(-, c) \xrightarrow{\theta} C(-, c \times c) \xrightarrow{\nu} C(-, c) $$

where $\nu = \mu_* = C(-, \mu)$, while the first natural isomorphism is given by the definition of the product object $c \times c$. Conversely, given any natural $\nu$, the Yoneda lemma proves that there is a unique $\mu : c \times c \rightarrow c$ such that $\nu = \mu_*$. A diagram chase shows that $\mu$ is associative only if $\bar{\mu}$ is.

Colimits of representatable functors

Any functor $K : D \rightarrow \textbf{Sets}$, from a small category $D$ to the category of Sets, can be represented, in a canoical way, by the colimit of a diagram of representable functors $\text{hom}(d, -)$, where $d \in D$. First, given $K$, we construct a diagram category for the colimit $J$, the "category of elements" of $K$; that is the comma category $1 \downarrow K$ of object pairs $(d, x)$, with $d \in D, x \in K(d)$ as objects, and arrows $f : (d, x) \rightarrow (d', x')$, those arrows $f: d \rightarrow d'$ for which $K(f)x = x'$. Then, $K$ is the colimit of the diagram on $1 \downarrow K$ given by the functor $$M : J^D \rightarrow \textbf{Sets}^D$$

which sends each object $(d, x)$ to the hom-functor $D(d, -)$, and each arrow $f$ to the induced NT $f_* : D(d', -) \rightarrow D(d, -)$. Then, the Yoneda isomorphism on $\textbf{Sets}^D$ yields a cone over base $M$ to $K$: $$y^{-1} : K(D) \rightarrow \text{Nat}(D(d, -), K)$$

and we get the following diagram:

$$ \begin{xy} \xymatrix{ J\ar[d]^M & (d, x)\ar[d] & (d', x')\ar[d]\ar[l]_{f_*} \\ \textbf{Sets}^D & D(d, -)\ar[d]_{y^{-1}x}\ar[dr]_{y^{-1}x'}|{\hole} & D(d', -)\ar[d]^{y^{-1}z'}\ar[dl]_{y^{-1}z}\ar[l]_{f_*} \\ & K\ar@{.>}[r]_\theta & L } \end{xy} $$

This cone to $K$ is colimiting over $D(d, -)$. Consider any other cone from $D(d, -)$ to the vertex $L$; by the Yoneda lemma, this is an NT $D \rightarrow \textbf{Sets}$, given by $y^{-1}z : D(d, -) \rightarrow L$, as well as by $y^{-1}z' : D(d', -) \rightarrow L$, for some $z \in L(D), fz = z'$. To show that the cone to $K$ is universal, construct an NT $\theta : K \rightarrow L$ and set $\theta_d x = z$. Since $f(y^{-1}z) = y^{-1}(fz) = y^{-1}z'$, $\theta$ is natural.

For small category $C$, the contravariant functor $F : C^{op} \rightarrow \textbf{Sets}$ is called a `presheaf`, and the category of these functors is written as $\hat{C}$. This terminology comes from the case when $C$ is a category of open subsets $U$ of a topological space, and $F$ smooth functions; then, for the inclusion $V \subset U$, we have the map $F(U) \rightarrow F(V)$.