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For small categories $\mathcal{C},\mathcal{D}$, a functor $F:\mathcal{C}\to \mathcal{D}$ and $Q:\mathcal{C}\to \mathbf{Set}$, choose an infinite cardinal number $\kappa$ bounding the number of morphisms of $\mathcal{D}$ and the total number of elements of $Q$. Given a natural transformation $\eta :Q\to GF$, generate a subfunctor $G^{\prime }\subseteq G$ from all ${\eta }_c(x)$ by applying every outgoing $\mathcal{D}$-morphism. Each $G^{\prime }(d)$ has at most $\kappa$ elements. Label bounded values by subsets of $\kappa$; only a set of such functors and transformations exists. These factorizations give the solution-set condition for precomposition, whose left adjoint is the left Kan extension.

A geometric morphism is essential when its inverse image has an additional left adjoint. For a functor of small categories, precomposition on presheaves has left Kan extension and Right Kan extension as its two adjoints. Precomposition preserves finite limits pointwise, so this is an essential geometric morphism.

A functor $F:\mathcal{C}\to \mathcal{D}$ induces a geometric morphism from the presheaf topos on $\mathcal{C}$ to that on $\mathcal{D}$. Its inverse image is precomposition with $F^{\mathrm{op}}$, its direct image is Right Kan extension and its extra left adjoint is left Kan extension. The extra left adjoint sends $yC$ to $y(FC)$.

A left Kan extension is left adjoint to precomposition, when it exists; a Right Kan extension is right adjoint. For small indexing categories and set-valued functors, both exist by comma-category colimit and limit formulas. These two adjoints give the essential geometric morphism induced by a functor between presheaf indexing categories.

For small categories and a set-valued functor $Q$, each displayed colimit exists in the Category of sets. A representative $(c,u:Fc\to d,x\in Qc)$ maps under a natural transformation $\eta :Q\to GF$ to $G(u){\eta }_c(x)$. The colimit identifications make this well-defined, proving the adjunction between left Kan extension and precomposition.

Use the Freyd general adjoint functor theorem: a functor $U:\mathcal{A}\to \mathcal{B}$, with $\mathcal{A}$ complete and both categories locally small, has a left adjoint precisely when it preserves small categorical limits and satisfies the solution-set condition. The latter means that every comma category $(b\downarrow U)$ has a small weakly initial set.

Here is the proof. First prove the initial-object lemma for complete categories with a weakly initial set. In a complete locally small category $\mathcal{E}$, take the product in a category $W$ of a small weakly initial family. It is weakly initial: project to whichever member admits a morphism into the required object. Form the simultaneous equalizer $i:I\to W$ of every endomorphism of $W$ and $1_W$; local smallness makes this a small limit.

For parallel $f,g:I\to X$, let $j:Y\to I$ be their equalizer. Weak initiality supplies $t:W\to Y$. Since $ijt$ is an endomorphism of $W$, the defining equalities give $ijti=i$. Cancel the monomorphism $i$ to get $jti=1_I$. Thus $j$ is a split epimorphism as well as a monomorphism, hence invertible, and $f=g$. Existence of a morphism $I\to X$ follows by composing $i$ with a map $W\to X$. Therefore $I$ is an initial object.

Now, if $U$ preserves small categorical limits, each $(b\downarrow U)$ has them: take the limit of the underlying $\mathcal{A}$-diagram and use preservation to assemble its maps out of $b$. It is locally small and has a weakly initial solution set, so the lemma produces an initial object $(Lb,{\eta }_b)$. Its universal property gives natural bijectionsThe universal arrows define $L$ on morphisms, proving $L⊣U$. Conversely, a right adjoint preserves categorical limits, and its unit supplies a singleton solution set in each comma category.

Apply this to $U=F^{\ast }$. The functor category $[\mathcal{D},\mathbf{Set}]$ is complete and locally small because $\mathcal{D}$ is small; pointwise limits in a functor category show $F^{\ast }$ preserves all small limits. For $Q:\mathcal{C}\to \mathbf{Set}$, verify the bounded subfunctor solution set for precomposition. Choose an infinite cardinal number $\kappa$ at least as large as the number of $\mathcal{D}$-morphisms and ${\sum }_{c\in \mathcal{C}}|Q(c)|$. Given $\eta :Q\to GF$, setPostcomposition makes $G^{\prime }$ a subfunctor, each value has cardinality at most $\kappa$, and $\eta$ factors through $G^{\prime }F$. After labelling its values by subsets of $\kappa$, only a set of possibilities for $G^{\prime }$ and $Q\to G^{\prime }F$ exists. Their maps into $G$ give the required weakly initial set. Thus $F^{\ast }$ has a left adjoint.

It is the left Kan extension. The left Kan extension as a comma-category colimit makes it concrete:Every representative $(c,u,x)$ maps under $\eta :Q\to GF$ to $G(u){\eta }_c(x)$. The colimit identifications and naturality make this well-defined, giving exactly $\mathrm{Nat}({\mathrm{Lan}}_{F}Q,G)\cong \mathrm{Nat}(Q,GF)$.

Both alternatives can be proved using the same smallness mechanism. We first establish the initial-object lemma for complete categories with a weakly initial set, then obtain the two adjoint functor existence theorems and their applications.

Let $\mathcal{E}$ be a locally small category with all small categorical limits and a weakly initial set $(W_i{)}_{i\in I}$. Its product in a category $W={\prod }_i{W}_i$ is weakly initial: given any object, choose a map from some $W_i$ and compose with the corresponding projection. Take the simultaneous equalizer $e:E\to W$ of every endomorphism of $W$ and its identity; local smallness makes this a small categorical limit. Then $E$ is also weakly initial. If $a,b:E\to X$, form their equalizer $j:Y\to E$ and choose $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, the construction of $e$ gives $ejte=e$. Cancelling the monomorphism $e$ yields $jte=1_E$. Thus $j$ is both a monomorphism and a split epimorphism, hence invertible, and $a=b$. There is therefore exactly one arrow from $E$ to every object: $E$ is initial.

The Freyd general adjoint functor theorem says that for a functor $U:\mathcal{A}\to \mathcal{B}$ between locally small categories, with $\mathcal{A}$ complete, a left adjoint exists exactly when $U$ preserves all small categorical limits and satisfies the solution-set condition. That condition requires, for each $B$, a set of arrows $u_i:B\to U{A}_i$ such that every $u:B\to UA$ factors as $U(f)u_i$ for some $i$ and $f:A_i\to A$.

For sufficiency, the comma category $(B\downarrow U)$ is locally small and complete: form a diagram's categorical limit in $\mathcal{A}$, and use categorical limit preservation to obtain its arrow from $B$. The solution set is a weakly initial set in this comma category. The lemma provides an initial object $(LB,{\eta }_B:B\to ULB)$. Its universal property givesFor $h:B\to B^{\prime }$, define $Lh$ by $U(Lh){\eta }_B={\eta }_{B^{\prime }}h$. Uniqueness proves the identity and composition laws and naturality of the displayed bijections, hence $L⊣U$. Conversely, an adjunction makes $U$ preserve categorical limits, since the corresponding hom-set functors turn a categorical limit cone into a categorical limit of sets. The single arrow ${\eta }_B$ is a solution set. This proves both directions of the theorem.

Now let $F:\mathcal{C}\to \mathcal{D}$ have small domain and codomain. The functor categories $[\mathcal{D},\mathbf{Set}]$ and $[\mathcal{C},\mathbf{Set}]$ are locally small, and their categorical limits are pointwise. Thus the precomposition functor $F^{\ast }$ preserves categorical limits and its domain is complete. To verify its solution-set condition, fix $X:\mathcal{C}\to \mathbf{Set}$ and an arrow $x:X\to F^{\ast }Y$. Let $Y_0\subseteq Y$ be the subfunctor generated by all elements $x_c(a)$, where $c\in \mathcal{C}$ and $a\in X(c)$. Explicitly,These sets are stable under every $Y(h)$. If $\kappa$ is an infinite cardinal bounding $|\mathrm{Mor}\mathcal{D}|$ and $\left|{\coprod }_{c}X(c)\right|$, every $Y_0(d)$ has cardinal at most $\kappa$. Replace each such set by a subset of a fixed set of size $\kappa$. There is only a set of choices of these subsets, of their action maps for the set of arrows of $\mathcal{D}$, and of maps from $X$ to their restrictions. These coded pairs form a solution set: $x$ factors through $F^{\ast }{Y}_0$ and then the inclusion $Y_0\hookrightarrow Y$. The general theorem now yieldsIts values may also be written ${\mathrm{Lan}}_{F}X(d)={\mathrm{colim}}_{(Fc\to d)}X(c)$, the usual left Kan extension formula.

For the other alternative, the categorical limit form of the Special adjoint functor theorem says that a limit-preserving functor $U:\mathcal{A}\to \mathcal{B}$ has a left adjoint if $\mathcal{A}$ is locally small, complete and well-powered, has a cogenerating set $(Q_i{)}_{i\in I}$, and $\mathcal{B}$ is locally small. A cogenerating set means that maps into its members distinguish any two parallel arrows. The dual theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family has a right adjoint. These conclusions are also necessary in their preservation clauses.

Here is a proof of the special theorem rather than an invocation of the general theorem without its required solution set. Given $(A,x:B\to UA)$, intersect all subobjects $m:M\hookrightarrow A$ through which $x$ factors after application of $U$. There are only a set of them by well-poweredness. Their intersection is a wide pullback, preserved by $U$, so gives a smallest supporting pair $(A_0,x_0)$. If $p,q:A_0\to Q_i$ satisfy $U(p)x_0=U(q)x_0$, their equalizer is a supporting subobject. Minimality makes that equalizer invertible, whence $p=q$. Consequentlyis injective. The cogenerating property makes the evaluation arrow from $A_0$ into the product in a category of the $Q_i$, one factor for each map $A_0\to Q_i$, a monic arrow. Its indexing set at each $i$ is identified with a subset of the fixed set $\mathcal{B}(B,U{Q}_i)$. Only a set of such products in a category can occur. Each has only a set of subobjects, and each possible subobject has only a set of arrows from $B$ into its image under $U$. These pairs give a weakly initial set in $(B\downarrow U)$, since every original pair receives the arrow from its minimal supporting pair. The complete-category lemma now gives the adjoint. Reversing all arrows proves the colimit form of the special theorem.

Apply that dual form to $F^{\ast }:[\mathcal{D},\mathbf{Set}]\to [\mathcal{C},\mathbf{Set}]$. Its domain is cocomplete and locally small, and $F^{\ast }$ preserves pointwise colimits. The set of covariant representable functors $\mathcal{D}(d,-)$ generates the domain: by the Yoneda lemma, maps from these representables probe every element at every object and hence distinguish distinct natural transformations. Finally, epimorphisms in this functor category are pointwise surjective. Indeed, their pointwise images form subfunctors; if an image is proper, the two inclusions into the pointwise pushout of two copies of the target over that image distinguish the original map, contradicting epicity. Quotients of a fixed functor are therefore specified by compatible equivalence relations on its sets of values, and form a set. Thus the domain is a well-copowered category. All special-theorem hypotheses hold, givingHere the Right Kan extension can be written ${\mathrm{Ran}}_{F}X(d)=\mathrm{Nat}(\mathcal{D}(d,F-),X)$. In particular both requested adjoints exist, although the two proofs use different smallness hypotheses.

Write $\hat{\mathcal{C}}=[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ and similarly for $\mathcal{D}$. The geometric morphism induced by a functor hasPrecomposition preserves all pointwise limits and colimits, so in particular it preserves finite limits. The Right Kan extension exists because the categories are small and sets have all small limits; its universal property gives $f^{\ast }⊣f_{\ast }$. Thus these functors define a geometric morphism $\hat{\mathcal{C}}\to \hat{\mathcal{D}}$.

There is also $f_{!}={\mathrm{Lan}}_{F^{\mathrm{op}}}$, a left Kan extension, with $f_{!}⊣f^{\ast }$. The Yoneda lemma identifies $f_{!}(yC)\cong y(FC)$, since for every $P$,This is the representable calculation used in the next parts.