Solution sets and universal representations

A functor may assign compatible data to every test object without offering a formula for a universal object. A solution set reduces that search to a small family. Colimits then combine the candidates, while their universal properties remove the ambiguity in choosing among them.

We fix universes, use small Hom sets, and take all families and diagram categories in the stated small universe. “All colimits” and “all limits” below mean all small ones. We assume the Yoneda convention from Ind-objects through their elements, the separate detecting and separating probes from Generators and small quotient families, and the intrinsic strict epimorphisms from Coimages, images and composition of quotients.

1. Universal elements and variance

Let $F:\mathcal C^{op}\to\operatorname{Set}$. Its category of elements $\mathcal E_F$ has objects $(X,x)$, where $x\in F(X)$, and arrows $\begin{gathered} (X,x)\xrightarrow{a}(Y,y),\\ a:X\to Y,\qquad F(a)(y)=x. \end{gathered} \tag{1.1}$ A universal element is a pair $(T,t)$ for which $\begin{gathered} \operatorname{Hom}_{\mathcal C}(X,T)\longrightarrow F(X),\\ a\longmapsto F(a)(t). \end{gathered} \tag{1.2}$ is bijective for every $X$. These functions are natural, since applying $F$ to $ab$ gives $F(b)F(a)$. Thus they represent $F$. The pair is universal precisely when $(T,t)$ is terminal in $\mathcal E_F$. This is the contravariant statement of Riehl, Proposition 2.4.8. For a covariant functor $H:\mathcal A\to\operatorname{Set}$, the corresponding object is initial and the formula uses $\operatorname{Hom}(T,X)$.

The variance affects preservation. A contravariant representation $\operatorname{Hom}(-,T)$ sends colimits in $\mathcal C$ to limits of sets. A covariant representation $\operatorname{Hom}(T,-)$ preserves limits in $\mathcal A$. Neither statement asserts preservation of colimits by a covariant representation.

Suppose $\mathcal C$ has all colimits and $F$ sends them to limits. Then the projection $\mathcal E_F\longrightarrow\mathcal C,\qquad (X,x)\longmapsto X \tag{1.3}$ creates colimits. Indeed, for a small diagram $(X_i,x_i)$, let $L=\operatorname{colim}_iX_i$, with structure maps $c_i$. Compatibility of the $x_i$ gives an element of $\lim_iF(X_i)$, so the preservation bijection supplies a unique $l\in F(L)$ with $F(c_i)(l)=x_i$. Given a cocone $a_i:(X_i,x_i)\to(Y,y)$, the underlying colimit supplies a unique $a:L\to Y$. The elements $F(a)(y)$ and $l$ agree after every $F(c_i)$, so are equal by that same bijection. Thus $a$ is a morphism of elements, proving its universal property.

For the empty diagram this says $F(0)=1$, and supplies an initial object of $\mathcal E_F$. In particular this category is nonempty. Its small coproducts and coequalizers also show that it is filtered: two objects map to their coproduct, and two parallel arrows are equalized by their coequalizer. Nothing here requires finite limits in $\mathcal C$.

2. A small solution set is enough

A solution set for $F$ is a small family of elements $(T_j,t_j)$ such that every $(X,x)\in\mathcal E_F$ has at least one arrow to a member of the family. The arrow need not be unique. This is a jointly weakly terminal set in $\mathcal E_F$.

A category is cofinally small if it receives a cofinal functor from a small category. We use the colimit convention: $K:\mathcal D\to\mathcal E$ is cofinal when every $e\downarrow K$ is nonempty and connected. It lets colimits over $\mathcal E$ be computed over $\mathcal D$ when the universal cocones in question exist.

Theorem 2.1. Let $\mathcal C$ have all colimits. For $F:\mathcal C^{op}\to\operatorname{Set}$, the following are equivalent:

$F$ is representable.
$F$ sends colimits to limits, and $\mathcal E_F$ has a solution set.
$F$ sends colimits to limits, and $\mathcal E_F$ is cofinally small.

The terminal-object construction below is the dual of the construction discussed in Riehl, Lemma 4.7.5. We give the complete colimit argument, including the uniqueness of the resulting arrows.

Proof. A representation makes $(T,t)$ terminal. Its singleton family is a solution set and the inclusion of that terminal object is cofinal. Preservation follows from the Hom universal property.

Assume preservation and a solution set, and let $\mathcal D\subset\mathcal E_F$ be the full subcategory on its objects. Local smallness makes $\mathcal D$ small, including all its morphisms. The comma category $e\downarrow\mathcal D$ is nonempty. Given two arrows $e\to d_1,e\to d_2$, form their pushout in $\mathcal E_F$; it has an arrow to some $d_3\in\mathcal D$. The resulting arrows $d_1\to d_3$ and $d_2\to d_3$ belong to $\mathcal D$ by fullness and agree on $e$. They connect the two comma objects. Thus this inclusion is cofinal. Conversely a small cofinal functor supplies a solution set by the nonemptiness of its comma categories. This proves the equivalence of the two smallness formulations under preservation.

We now construct the terminal object. Let $T=\operatorname{colim}_{d\in\mathcal D}d$ in $\mathcal E_F$, with maps $\eta_d:d\to T$. For any $e$, choose $h:e\to d$ and put $\lambda_e=\eta_dh$. This is independent of the choice: push out two choices over $e$, then map that pushout to a member $d_3$ of the solution set. The two composites to $T$ agree by the colimit cocone equations for the resulting maps in the full $\mathcal D$.

For $f:e\to e'$, a choice $e'\to d$ also gives a choice $e\to d$, so $\lambda_{e'}f=\lambda_e$. For $d\in\mathcal D$, choosing its identity gives $\lambda_d=\eta_d$. Therefore $\lambda_T\eta_d=\eta_d$ for every $d$, and colimit uniqueness gives $\lambda_T=\operatorname{id}_T$. For any arrow $a:e\to T$, naturality now gives $a=\lambda_Ta=\lambda_e$. This makes $T$ terminal. Its underlying object and chosen element represent $F$ by §1. $\square$

The full small subcategory must include its morphisms. A coproduct of a solution family alone is merely weakly terminal; the cocone relations in the colimit enforce uniqueness.

3. Co-wellpowered categories and the detecting-generator interface

An ordinary quotient of $X$ is an epimorphism $q:X\to Q$, considered up to isomorphisms of targets that commute with the specified map from $X$. Call $\mathcal C$ co-wellpowered if the ordinary quotient classes of each $X$ form a small set. This includes all epimorphisms, whether or not they are strict.

The role of ordinary quotient-smallness is to bound the possible targets of one fixed presentation object. This gives a direct solution-set proof of the representability principle, without assuming finite limits. Compare the dual special adjoint functor theorem in Riehl, Theorem 4.7.10.

Theorem 3.1. If $\mathcal C$ is cocomplete, co-wellpowered and has a small separating family, every $F:\mathcal C^{op}\to\operatorname{Set}$ sending colimits to limits is representable. The conclusion also holds if the family is only assumed detecting.

Proof for a separating family. For each object $X$, the evaluation map $e_X:P_X\to X$ from (3.3) is epic: two maps agreeing after it agree after every probe $G_i\to X$, hence agree by separation. Define a fixed small coproduct $Z_0=\coprod_{\substack{i\in I\\a\in F(G_i)}}G_i.$ Preservation gives $z_0\in F(Z_0)$ whose component at $(i,a)$ is $a$. For $(X,x)$, send the summand of $P_X$ indexed by $(i,b:G_i\to X)$ to the summand of $Z_0$ indexed by $(i,F(b)(x))$. This defines $h:P_X\to Z_0$, and the componentwise preservation bijection gives $F(h)(z_0)=F(e_X)(x).$ Form the pushout $X'=X\amalg_{P_X}Z_0$, with maps $c:X\to X'$ and $d:Z_0\to X'$. Preservation of this pushout supplies $x'\in F(X')$ with $F(c)(x')=x$ and $F(d)(x')=z_0$. Thus $(X,x)\to(X',x')$ is a morphism of elements. Also $d$ is epic. Indeed, two maps out of $X'$ agreeing after $d$ agree after $ce_X=dh$; cancel $e_X$ to make them agree after $c$, and use the pushout property to equate them.

Choose a small representative family of ordinary quotients of $Z_0$. Decorate each representative target $Q$ by every element of the small set $F(Q)$. This is a small family of elements. Each $(X',x')$ above is isomorphic over its quotient map to one of these decorated targets, so every $(X,x)$ maps to a member of the family. It is a solution set. Theorem 2.1 now gives a representing object $T$ and natural bijections $\operatorname{Hom}_{\mathcal C}(X,T)\simeq F(X). \tag{3.1}$ The construction includes an empty separating family: the fixed coproduct is then initial, and the same cancellation and pushout argument applies.

Bridge for a detecting family. We first prove that cocompleteness and co-wellpoweredness give an epi–mono factorization of every $f:P\to X$. Choose a small set of representatives $e_i:P\to Q_i$ of all ordinary quotient classes through which $f$ factors. Write $f=v_ie_i$; each $v_i$ is unique because $e_i$ is epic. The identity quotient is among the classes, so the set is nonempty.

Form the colimit of the small star diagram consisting of $P$ and all the arrows $e_i$. It gives maps $\begin{gathered} q:P\to Q,\qquad r_i:Q_i\to Q,\\ r_ie_i=q. \end{gathered} \tag{3.2}$ The maps $f,v_i$ form a cocone, giving $v:Q\to X$ with $vq=f$. The map $q$ is epic: if two maps out of $Q$ agree after $q$, their composites with every $r_i$ agree after the epic $e_i$, hence agree, and the colimit property makes the two maps equal.

To prove $v$ monic, suppose $a,b:W\to Q$ satisfy $va=vb$. Form their coequalizer $c:Q\to Q'$. It is epic, and $v=v'c$ for some $v'$. Thus $cq:P\to Q'$ is an ordinary quotient through which $f$ factors. It has the class of some $e_k$, so there is an isomorphism $s:Q_k\to Q'$ with $se_k=cq$. Put $d=r_ks^{-1}:Q'\to Q$. Then $dcq=r_ke_k=q.$ Cancel $q$ to get $dc=\operatorname{id}_Q$, and cancel $cq$ in $cdcq=cq$ to get $cd=\operatorname{id}_{Q'}$. Hence $c$ is invertible; since $ca=cb$, we have $a=b$. This proves the factorization.

Now let $(G_i)_{i\in I}$ be a small detecting family and use the canonical evaluation map $e_X:\coprod_{(i,a),\ a:G_i\to X}G_i\longrightarrow X. \tag{3.3}$ Factor $e_X=vq$ by the result just proved, with $q$ epic and $v$ monic. For every $i$, $\operatorname{Hom}(G_i,v)$ is injective by monicity. It is surjective because each $a:G_i\to X$ lifts to its summand of the evaluation coproduct and then through $q$. Detection makes $v$ invertible. Hence $e_X$ is epic.

If maps $b,c:X\to Y$ agree after every map from every $G_i$, they agree after $e_X$, so $b=c$. The detecting family is consequently separating, and the solution-set proof above applies. This also checks the empty-family case whenever it is detecting. No finite-limit hypothesis was introduced. $\square$

For a single generator $G$, the fixed object used in the proof is simply $Z_0=\coprod_{a\in F(G)}G. \tag{3.4}$ Thus the solution set uses quotients of one explicitly specified object, rather than coproducts depending on every possible $X$.

The quotient-smallness assumption is used here for ordinary quotients of $Z_0$. Smallness only for strict quotients does not justify dropping it. The next section treats the case in which all ordinary epimorphisms are strict.

4. Strict quotients, limits and right adjoints

Corollary 4.1. Suppose $\mathcal C$ is cocomplete, has finite limits and a detecting generator, and every epimorphism is strict. Then $F:\mathcal C^{op}\to\operatorname{Set}$ is representable exactly when it sends colimits to limits.

Proof. The kernel-pair argument in Generators and small quotient families, §4 gives a small set of strict quotient classes of each object. Since all epimorphisms are strict, these are all ordinary quotient classes. Thus $\mathcal C$ is co-wellpowered, and Theorem 3.1 applies. Necessity follows from the Hom universal property. $\square$

This applies to sets and to left modules over any unital ring. In sets the singleton is detecting; in left modules the regular module $R$ is detecting by evaluation at $1$. Both categories have the required colimits and finite limits. All their maps are strict by the explicit Set and arbitrary-ring module calculations in Coimages, images and composition of quotients, §5. Small module colimits are formed by a direct sum modulo the relations imposed by diagram arrows: a map out is precisely a compatible family of linear maps. Small limits are the compatible tuples in a product. These descriptions apply without commutativity or finite-generation assumptions.

The representability principle has consequences beyond an individual functor.

Theorem 4.2. Suppose $\mathcal C$ is cocomplete and every $F:\mathcal C^{op}\to\operatorname{Set}$ sending colimits to limits is representable. Then:

$\mathcal C$ has all limits.
For any locally small $\mathcal D$, a functor $U:\mathcal C\to\mathcal D$ has a right adjoint exactly when it preserves colimits.

Proof. For a small diagram $A:I\to\mathcal C$, define $H(X)=\lim_{i\in I}\operatorname{Hom}(X,A_i). \tag{4.1}$ This is the set of cones with vertex $X$. For a small diagram $X_j$ with colimit $X$, the Hom universal property gives $\begin{gathered} H(X)=\lim_i\lim_j\operatorname{Hom}(X_j,A_i)\\ \simeq\lim_j\lim_i\operatorname{Hom}(X_j,A_i)\\ =\lim_jH(X_j). \end{gathered} \tag{4.2}$ The interchange bijection sends a compatible family of compatible tuples to the same doubly indexed family in the other order. This covers arbitrary small diagram categories, including empty ones. The representing object for $H$ has the required cone universal property, so is a limit of $A$.

Now suppose $U$ preserves colimits. For each $Y\in\mathcal D$, the functor $H_Y(X)=\operatorname{Hom}_{\mathcal D}(U(X),Y) \tag{4.3}$ sends colimits to limits, so choose a representing object $R(Y)$ with its natural identification. A map $b:Y\to Y'$ gives, by postcomposition, a natural transformation $H_Y\to H_{Y'}$. Yoneda determines a unique $R(b):R(Y)\to R(Y')$ implementing it. Identity and composite transformations determine identity and composite maps, so $R$ is a functor. Its representing bijections give $\begin{gathered} \operatorname{Hom}_{\mathcal D}(U(X),Y)\\ \simeq\operatorname{Hom}_{\mathcal C}(X,R(Y)). \end{gathered} \tag{4.4}$ natural in both variables by construction. Thus $U\dashv R$.

Conversely, if $U\dashv R$, apply (4.4) to a colimit of $X_j$. The Hom universal property in $\mathcal C$ identifies the right side with $\lim_j\operatorname{Hom}(X_j,R(Y))$, hence the left side with $\lim_j\operatorname{Hom}(U(X_j),Y)$, naturally in $Y$. This says exactly that $U(\operatorname{colim}_jX_j)$, with its image cocone, is a colimit of the $U(X_j)$. No general completeness or cocompleteness hypothesis on $\mathcal D$ is required. $\square$

The construction of $R$ includes its action on arrows and both naturality variables. Choosing unrelated representatives on objects without their universal identifications would not by itself define an adjoint.

5. Covariant set functors

The solution-set construction also solves a covariant problem. Here it is necessary to bound the size of a carrier of each element.

Theorem 5.1. Every functor $H:\operatorname{Set}\to\operatorname{Set}$ preserving all small limits is naturally isomorphic to $X\mapsto X^A$ for some small set $A$.

Proof. Since $H$ preserves equalizers, it preserves injections. Indeed an injection of sets is isomorphic to an equalizer of two maps: take its image and the characteristic function of that image, equalized with the constant-one function. Applying $H$ preserves this equalizer and hence its monomorphism.

Given $x\in H(X)$, consider all subsets $D\subseteq X$ through whose inclusion $x$ lifts. There is at least one, namely $X$. Each lift is unique by preservation of injections. The diagram of these inclusions has their intersection $A$ as its limit. Its corresponding $H$-diagram has the compatible tuple consisting of $x$ and its lifts. Limit preservation therefore supplies $a\in H(A)$ mapping to $x$. This $A$ is the smallest subset through which $x$ lifts.

Put $2=\{0,1\}$. The map $\begin{gathered} \operatorname{Map}(A,2)\longrightarrow H(2),\\ f\longmapsto H(f)(a) \end{gathered}$ is injective. For if $H(f)(a)=H(g)(a)$, preservation of the equalizer $E\hookrightarrow A$ of $f,g$ makes $a$ lift through $E$. Then $x$ lifts through the subset $E\subseteq X$, so minimality gives $E=A$, hence $f=g$.

Every such $A$ embeds in the fixed small set $B=2^{H(2)}$. Explicitly, for $u\in A$, define $b_u:H(2)\to2$ by $b_u(H(f)(a))=f(u),$ and give it value zero outside this injective map's image. Distinct $u,v$ are distinguished by a function $f:A\to2$, so $u\mapsto b_u$ is injective. Transport $a$ to $H(C)$, where $C\subseteq B$ is the image of this embedding. The resulting decorated subset $(C,c)$ maps to $(X,x)$. Therefore the family of all subsets $C\subseteq B$, each decorated by all elements of $H(C)$, is a small weakly initial family in the category of elements of $H$.

That category is complete: for a small diagram of decorated sets, preservation of its underlying limit gives the unique compatible decoration and the limit universal property, exactly as in §1 with arrows reversed. It is locally small. Apply the terminal-object construction in the proof of Theorem 2.1 with all arrows reversed. In detail, take the full small subcategory on the weakly initial family and form its limit. For any target element, choose a map from a family member; the limit projection followed by that map is independent of the choice, since a pullback of two choices receives a map from another family member. These maps are natural in the target. At the limit itself the induced endomorphism agrees with the identity after every limit projection, hence is the identity. Naturality then makes every map from the limit to a target equal to the constructed one. Thus this decorated limit is initial.

Write it as $(A_0,a_0)$. By §1 its universal maps give natural bijections $\begin{gathered} \operatorname{Hom}(A_0,X)\longrightarrow H(X),\\ f\longmapsto H(f)(a_0). \end{gathered} \tag{5.1}$ This proves the power-functor assertion, including empty carriers. Conversely every such power functor preserves limits by the covariant Hom universal property. The representing set is unique up to a unique isomorphism preserving the chosen representation. Exercise 3 determines all natural transformations between these functors; Exercise 4 shows why finite limits do not suffice. $\square$

6. Graded exercises with complete solutions

Exercise 1 (warm-up: two distinguished subsets). For a set $X$, let $F(X)$ be the set of ordered pairs $(A,B)$ of disjoint subsets of $X$. Define its action on maps by inverse image. Find a terminal object of its category of elements and an explicit representation. What changes if disjointness is omitted?

Solution. Inverse images preserve disjointness, identities and composition, so this is a contravariant functor. Let $T=\{a,b,c\}$ with distinguished pair $(\{a\},\{b\})$. For any disjoint pair $(A,B)$ in $X$, the unique function to $T$ with these inverse images assigns $a$ on $A$, $b$ on $B$, and $c$ elsewhere. These are all the points, since the subsets are disjoint. The inverse construction takes a function's inverse images of $\{a\},\{b\}$. These operations are inverse and commute with inverse image along every $X'\to X$. Thus $F\simeq\operatorname{Hom}(-,T)$ and the displayed element is terminal.

If arbitrary pairs are allowed, use $T'=2\times2$ and its two coordinate-one subsets. A point maps to its two membership bits, including $(1,1)$ for points in the intersection. Taking coordinate inverse images reverses this construction. The representative has four points instead of three. Both functors send all colimits to limits by their explicit representations.

Exercise 2 (standard: open sets and ordinary quotients). Let $\mathcal O(X)$ be the set of open subsets of a topological space, with inverse image along continuous maps. Represent $\mathcal O:\operatorname{Top}^{op}\to\operatorname{Set}$. Then verify that all topological spaces satisfy the separating-family form of Theorem 3.1, although the singleton does not detect isomorphisms.

Solution. Let $S=\{0,1\}$ have open sets $\varnothing,\{1\},S$. A function $\chi_U:X\to S$ is continuous exactly when $U=\chi_U^{-1}(\{1\})$ is open, because the other two inverse images are automatically open. Thus $\operatorname{Hom}_{\operatorname{Top}}(X,S)\simeq\mathcal O(X)$, naturally under inverse image, with universal open set $\{1\}$. In particular this functor sends all colimits to limits.

The singleton is separating in $\operatorname{Top}$: its maps pick every point, so equality after all these maps is equality of underlying functions. It is not detecting, as the continuous bijection from a discrete two-point space to an indiscrete one is not a homeomorphism.

To check cocompleteness, take the colimit of the underlying sets of a small diagram and give it the final topology for the maps from its spaces: a subset is open exactly when its inverse image in every diagram object is open. This is a topology, and a function out is continuous exactly when all its composites with the diagram maps are continuous. This proves the required colimit property, including the empty space for the empty diagram.

For co-wellpoweredness, epimorphisms in all of $\operatorname{Top}$ are exactly surjections, as proved in the prerequisite strict-morphism lesson. An ordinary quotient $q:X\to Y$ is encoded by its fibre equivalence relation on the set $X$, together with the topology transported to the fibre set $X/{\sim_q}$ by $[x]\mapsto q(x)$. There is a small set of equivalence relations on $X$, and a small set of topologies on each of their quotient sets, since topologies are subsets of the power set. Equal codes give the unique compatible homeomorphism of targets. Thus all ordinary quotient classes form a small set, even though many of these quotients are not strict. Local smallness is immediate from the set of underlying functions.

All hypotheses of the separating-family form of Theorem 3.1 hold. It follows that every contravariant set-valued functor on all topological spaces sending colimits to limits is representable. This conclusion uses the separating singleton and co-wellpoweredness; it does not replace them by a detecting-singleton assertion.

Exercise 3 (advanced: all natural operations on power functors). For small sets $A,B$, determine every natural transformation $X^A\to X^B$. Explain the direction of the resulting function between $A$ and $B$, and classify the natural endomorphisms of the identity functor on sets.

Solution. Write $P_A(X)=\operatorname{Hom}(A,X)$. For a natural transformation $\theta:P_A\to P_B$, evaluate at the identity: $s=\theta_A(\operatorname{id}_A)\in\operatorname{Hom}(B,A).$ For $f:A\to X$, naturality along $f$ says $\theta_X(f)=f\circ s. \tag{6.1}$ Thus $\theta$ is determined by the function $s:B\to A$. Conversely precomposition with any such $s$ defines a natural transformation by associativity of composition. Evaluation at the identity recovers $s$, proving $\operatorname{Nat}(P_A,P_B)\simeq\operatorname{Hom}(B,A). \tag{6.2}$ The direction reverses because these functors are covariant represented functors. For a $B$-coordinate $b$, the output reads the $A$-coordinate $s(b)$, allowing repeated or omitted coordinates.

The identity functor is $P_1$, so its natural endomorphisms correspond to the unique function $1\to1$. Its only natural endomorphism is the identity. Empty cases are included: $P_{\varnothing}$ is the constant singleton functor, and there is no natural map from it to the identity, because there is no function $1\to\varnothing$. There is a unique natural map from the identity to the constant singleton functor.

Exercise 4 (expert: finite limits do not give a representation). Let $H(X)=X^{\mathbb N}/{\sim}, \tag{6.3}$ where sequences are equivalent if their entries are equal for all sufficiently large $n$. Use componentwise application of functions to make $H$ a functor. Prove that it preserves finite limits but not countable products, and hence is not representable.

Solution. Eventual equality is an equivalence relation and is preserved by applying a function, so the action is well-defined. Componentwise identities and composition give the functor laws.

There is exactly one sequence in a singleton set, so $H(1)=1$. The comparison $H(X\times Y)\longrightarrow H(X)\times H(Y)$ is surjective: pair representatives of the two classes term by term. It is injective: eventual equality in each coordinate implies eventual equality of pairs after the maximum of the two bounds. Thus it preserves binary products, and by induction all finite products including the empty one.

For $a,b:X\to Y$, let $E=\{x\in X:a(x)=b(x)\}$. The map $H(E)\to H(X)$ is injective, since equality of sequences in $E$ is tested by the same eventual relation in $X$. Its image lies in the equalizer of $H(a),H(b)$. Conversely a class $[x_n]$ in that equalizer has $a(x_n)=b(x_n)$ for every $n\ge N$, so its tail lies in $E$. In particular $E$ is nonempty; replace the finitely many earlier entries by $x_N\in E$. This gives an $E$-sequence with the same class. If $E$ is empty, no such equalizer class exists and $H(E)$ is empty as required. Thus $H$ preserves equalizers. Finite products and equalizers construct every finite limit by taking compatible tuples, so it preserves finite limits.

For the product of countably many copies of $2$, put $X=2^{\mathbb N}$. Let $z_n\in X$ be the all-zero tuple, and let $e_n\in X$ have value $1$ at coordinate $n$ and $0$ elsewhere. The sequence classes $[z_n]$ and $[e_n]$ in $H(X)$ are distinct: their entries differ at every $n$. But for each fixed coordinate $j$, the two coordinate sequences differ only when $n=j$, so their classes in $H(2)$ are equal. Consequently $H(2^{\mathbb N})\longrightarrow H(2)^{\mathbb N} \tag{6.4}$ is not injective. It fails to preserve this countable product. Every covariant represented functor preserves all limits, so $H$ is not representable. This identifies the exact missing hypothesis rather than contradicting Theorem 5.1.

Original lesson text: CC0 1.0. Adjoint and universal-element results have complete proofs in this lesson and its specified prerequisites; cited references give mathematical credit, and no source prose is reproduced. Self-checked; no independent review has occurred.