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

## 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:

- \(\textbf{Vec}_K\) 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.
- 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.
- 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](/ct/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 \(F(U) \rightarrow F(V)\).