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The general adjoint functor theorem has the following limit form. Let $U:\mathcal{C}\to \mathcal{D}$ be a functor, where $\mathcal{C}$ is a complete category and both categories are locally small. Then $U$ has a left adjoint exactly when it preserves small limits and satisfies the solution-set condition.

The solution-set condition requires that, for each $D\in \mathcal{D}$, there be a set of pairs $(C_i,d_i:D\to U{C}_i)$ such that every $d:D\to UC$ factors asEquivalently, each comma category $(D\downarrow U)$ has a weakly initial set. All completeness and preservation requirements here concern small categorical limits. Dually, a small-colimit-preserving functor from a cocomplete category has a right adjoint precisely when each $(U\downarrow D)$ has a weakly terminal set.

The Special adjoint functor theorem says: if $\mathcal{C}$ is a locally small category, a complete category, a well-powered category, and has a small cogenerating family, then a functor $U:\mathcal{C}\to \mathcal{D}$ into any locally small category has a left adjoint if and only if it preserves small categorical limits.

A small cogenerating family $(Q_i{)}_{i\in I}$, with $I$ a set, distinguishes unequal parallel morphisms by postcomposition: for $f\ne g:A\to B$, some $q:B\to Q_i$ has $qf\ne qg$. Being well-powered means that the subobjects of each object form a set up to the usual equivalence of monomorphisms. These hypotheses make the solution-set condition automatic.

The dual formulation uses a cocomplete category, being a well-copowered category and having a small generating family, and concludes that every small-colimit-preserving functor has a right adjoint.

For a locally small category $\mathcal{C}$, a representation of a functor $F:\mathcal{C}\to \mathbf{Set}$ is an object $A$ and an element $a\in F(A)$ such thatis a bijection for every $B$, naturally in $B$. Equivalently, $\theta :\mathcal{C}(A,-)\cong F$ is a natural isomorphism.

Suppose $(A,a)$ and $(A^{\prime },a^{\prime })$ are two representations. Universality gives unique mapssuch that $F(u)(a)=a^{\prime }$ and $F(v)(a^{\prime })=a$. Then $F(vu)(a)=a$, and uniqueness applied to the element $a$ gives $vu=1_A$. Similarly $uv=1_{A^{\prime }}$. Thus the representing objects are uniquely isomorphic in a way carrying one universal element of a set-valued functor to the other.

For $F:\mathcal{C}\to \mathcal{D}$ and $B\in \mathcal{D}$, the comma category $(B\downarrow F)$ has objects $(A,u)$ with $u:B\to F(A)$. A morphism $(A,u)\to (A^{\prime },u^{\prime })$ is a map $h:A\to A^{\prime }$ satisfyingThe universal arrow from an object to a functor criterion says that $F$ has a left adjoint exactly when $(B\downarrow F)$ has an initial object for every $B$. Indeed, an initial $(LB,{\eta }_B)$ represents the functor $A\mapsto \mathcal{D}(B,FA)$, and uniqueness makes $B\mapsto LB$ functorial.

When $F:\mathcal{C}\to \mathbf{Set}$ and $1$ is a singleton, an arrow $1\to F(A)$ is just an element $a\in F(A)$. Hence $(1\downarrow F)$ is the category of elements, and its initial objects are exactly the representations of $F$. This proves

If $F$ has a left adjoint, the universal-arrow criterion immediately makes it representable. Conversely, suppose $\mathcal{C}$ is cocomplete and $F\cong \mathcal{C}(A,-)$. For a set $S$, form the copowerThe coproduct in a category universal property gives natural bijectionsso $L⊣F$.

Cocompleteness cannot be omitted. Let $\mathcal{C}$ be the category of ordinals in reverse order: there is one arrow $\alpha \to \beta$ exactly when $\alpha \ge \beta$ in the ordinary ordering. This large poset is locally small and complete. For a set-indexed family $({\alpha }_i)$, its product in the reversed order is the ordinary supremum ${\sup }_i{\alpha }_i$, and equalizers in a poset are automatic. But $\mathcal{C}$ has no initial object, since that would be a largest ordinal. Any representable functor $\mathcal{C}(A,-)$ is therefore the requested example: if it had a left adjoint $L$, thenwould be a singleton for every $B$, making $L(\varnothing )$ initial, a contradiction.