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We use the general adjoint functor theorem in its solution-set form. Let $G:\mathcal{D}\to \mathcal{C}$ be a functor between locally small categories, with $\mathcal{D}$ a complete category. Then $G$ has a left adjoint exactly when it preserves all small categorical limits and, for each $C\in \mathcal{C}$, there is a set-indexed familysuch that every $u:C\to GD$ factors as $Gf{u}_i$ for some $i$ and some $f:D_i\to D$. This is the solution-set condition. The indexing family is a set, whereas the entire comma category $(C\downarrow G)$ can be large.

We first prove the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{E}$ be locally small and complete, with a weakly initial set $(E_i{)}_{i\in I}$. Its product in a category $W={\prod }_i{E}_i$ is weakly initial: follow a projection by a map from the corresponding $E_i$ to any desired object. The hom-set $\mathcal{E}(W,W)$ is a set, so we can form the simultaneous equalizerThis is a small limit; equivalently, equalize the family of endomorphisms with the family of identity maps into a product of copies of $W$. Every object receives a map from $E$, since it receives one from $W$.

To prove uniqueness, let $a,b:E\rightrightarrows B$, and take their equalizer $j:V\to E$. Weak initiality of $W$ gives $t:W\to V$. Since $ejt$ is an endomorphism of $W$, the defining property of $e$ givesThe monomorphism $e$ cancels, yielding $jte=1_E$. Thus $j$ is both a monomorphism and a split epimorphism, hence an isomorphism. Its equalizer property now implies $a=b$. Therefore $E$ is initial.

Fix $C$. Small categorical limits in $(C\downarrow G)$ are obtained from the limits in $\mathcal{D}$: preservation by $G$ supplies the unique map from $C$ to the image of the limiting object. The comma category is locally small, and the solution family is a weakly initial set in it. The lemma supplies an initial object $(LC,{\eta }_C:C\to GLC)$.

For $k:C\to C^{\prime }$, initiality gives a unique arrow $Lk:LC\to L{C}^{\prime }$ satisfying $G(Lk){\eta }_C={\eta }_{C^{\prime }}k$. Uniqueness proves preservation of identities and composition, so $L$ is a functor. Initiality also gives natural bijectionswhich prove $L⊣G$.

Conversely, if $L⊣G$, the singleton ${\eta }_C:C\to GLC$ is a solution set by the adjunction bijection. To see that $G$ preserves a small limit $D={\lim }_j{D}_j$, use the adjunction and the limit property to obtain, naturally in $C$,This is precisely the universal property of the image limit cone. It proves the necessity of both hypotheses.

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.

Definitions organize category theory, but theorems give that organization mathematical force. I would therefore argue against treating the subject as an exception in which definitions are intrinsically more important than theorems. The distinction is especially misleading here because good categorical definitions are designed to expose hypotheses from which substantial theorems follow.

A category records objects, composable arrows, identities and associative composition. A functor preserves that structure. This language allows one to compare otherwise different mathematical settings, but it does not itself say which constructions exist. For example, the definition of a product in a category is not a recipe for forming a Cartesian product of underlying sets: it requires a universal factorization of pairs of arrows. If two products in a category $(P,p_1,p_2)$ and $(Q,q_1,q_2)$ exist, the universal properties give unique arrows $f:P\to Q$ and $g:Q\to P$ respecting projections. The composites $gf$ and $fg$ respect the same projections as the identities, so uniqueness forces them to be identities. Thus a definition yields a genuine theorem: products in a category are unique up to unique compatible isomorphism. The proof matters because it licenses replacing a construction by any other realization of the same property.

The Yoneda lemma is a more substantial example. For a covariant representable functor $\mathcal{C}(A,-)$ and $X:\mathcal{C}\to \mathbf{Set}$, a natural transformation $t:\mathcal{C}(A,-)\to X$ is determined by $a=t_A(1_A)$. Naturality forcesConversely this formula defines a natural transformation for every $a\in X(A)$. Hence $\mathrm{Nat}(\mathcal{C}(A,-),X)\cong X(A)$. The definition of naturality is indispensable, but the theorem identifies every possible transformation, proves representables detect arrows, and explains why universal objects can be studied through hom-sets. Merely knowing the definition does not supply this conclusion.

Similarly, an adjunction is defined by natural hom-set bijections. The construction of a free abelian group gives $\mathbf{Ab}({\mathbb{Z}}^{(S)},A)\cong \mathbf{Set}(S,UA)$, because a group homomorphism is determined uniquely by its values on the basis. This makes a universal extension problem transparent. But the Freyd general adjoint functor theorem goes further: it turns categorical limit preservation and a solution set into an actual adjoint. Its smallness hypotheses cannot be discarded as bookkeeping. For instance, the underlying-set functor from finite groups to finite sets has no left adjoint at a singleton. A putative universal finite group with universal element of order $m$ could not map that element to a generator of a finite cyclic group of order greater than $m$. The theorem's hypotheses separate existence from attractive notation.

The monoidal coherence theorem offers another example. A monoidal category has associativity and unit isomorphisms satisfying the pentagon identity for a monoidal category and triangle identity for a monoidal category. Coherence proves that every formal reassociation and unit insertion produces the same structural map. This is what justifies calculating without displaying all parentheses; it is not true merely because one has named an associator. Nor does it identify arbitrary additional maps, such as distinct braidings.

Finally, the opposite category construction shows how a definition can economize on proofs: reversing arrows takes products in a category to coproducts in a category, initial objects to terminal objects and monads to comonads. Yet this economy works because theorems and their proofs are invariant under the reversal. The subject is powerful precisely when definitions isolate reusable structure and theorems establish its consequences. Separating their importance obscures that partnership.

Take $A=1$. By assumption $E=(1\downarrow F)$ has finite limits, so $E^{\mathrm{op}}$ has finite colimits. A filtered category is nonempty, admits a cocone over every pair of objects, and admits a coequalizing arrow for every parallel pair. The initial object of $E^{\mathrm{op}}$ gives nonemptiness, its binary coproducts give the required two-object cocones, and its coequalizers equalize parallel arrows. Consequently $\boxed{\text{(ii)}\Rightarrow \text{(iii)}}$. These statements also handle the otherwise possible empty category of elements.

The general adjoint functor theorem says that, for a functor $U:\mathcal{A}\to \mathcal{B}$ between locally small categories, with $\mathcal{A}$ a complete category, $U$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition. The solution-set condition means that for each $B\in \mathcal{B}$ there is a set of arrows $b_i:B\to U{A}_i$ such that every $b:B\to UA$ factors as $U(a)b_i$ for some $i$ and some $a:A_i\to A$.

Here is the smallness argument behind the theorem. A complete category that is locally small and has a weakly initial set $(W_i)$ has an initial object. Form the product in a category $W={\prod }_i{W}_i$. It is weakly initial, since a projection $W\to W_i$ followed by a chosen arrow $W_i\to X$ gives an arrow $W\to X$. Take the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ with $1_W$; the family is a set by local smallness. The object $E$ is also weakly initial. For parallel arrows $a,b:E\rightrightarrows X$, take their equalizer $j:Y\to E$ and a map $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, the defining property of $e$ gives $ejte=e$. Cancellation of the monomorphism $e$ gives $jte=1_E$. Thus the monomorphism $j$ has a right inverse and is an isomorphism, so $a=b$. Existence of arrows out of $E$ was already established, proving the initial-object lemma for complete categories with a weakly initial set.

If $U$ preserves small categorical limits, the comma category $(B\downarrow U)$ is complete: take the limiting object in $\mathcal{A}$, and use preservation to assemble the arrows from $B$. It is locally small, and the solution-set condition provides a weakly initial set. Its initial object is an arrow ${\eta }_B:B\to ULB$ with a natural bijectionFor a morphism $v:B\to B^{\prime }$, define $Lv$ by $U(Lv){\eta }_B={\eta }_{B^{\prime }}v$. Uniqueness proves functoriality and naturality, giving a left adjoint $L⊣U$. Conversely, a right adjoint preserves categorical limits, because applying the adjunction converts a limiting cone over a diagram into the corresponding limiting cone of hom-sets. Its adjunction unit at $B$ is a singleton solution set. This proves both directions of the general adjoint functor theorem.

Apply it to the inclusion $J:\mathbf{KHaus}\hookrightarrow \mathbf{Top}$, where $\mathbf{KHaus}$ is the full subcategory of compact Hausdorff spaces. A small product in a category of compact Hausdorff spaces is again compact Hausdorff by the Tychonoff theorem. An equalizer of two continuous functions into a Hausdorff space is closed, hence compact Hausdorff. Products and equalizers construct all small categorical limits, and the inclusion preserves these constructions. Both categories are locally small.

Fix a topological space $X$ and a continuous function $f:X\to K$ into a compact Hausdorff space. The closure $K_f=\overline{f(X)}$ is compact Hausdorff and has a dense subset of cardinality at most $|X|$. The permitted cardinal estimate givesFor $X=\varnothing$, use $K_f=\varnothing$. Choose a set of representatives of all compact Hausdorff topologies on underlying sets of cardinality at most this bound, and all continuous functions from $X$ to those representatives. Every $f$ factors through one of them, using the inclusion $K_f\hookrightarrow K$. This is the solution-set condition. Consequently the inclusion of compact Hausdorff spaces has a left adjoint, the compact Hausdorff reflection:The universal arrow $X\to RX$ has dense image: factoring it through its closed image and applying uniqueness supplies an inverse to that image inclusion. It need not be an embedding when $X$ is not sufficiently separated; the assertion is a compact Hausdorff reflection, rather than a claim that every topological space embeds in a compact Hausdorff space.

For completeness, the alternative Special adjoint functor theorem can be proved from the same initial-object argument. Its limit form assumes that $\mathcal{A}$ is complete, locally small and well-powered, with a small cogenerating family $(Q_i{)}_{i\in I}$; then $U:\mathcal{A}\to \mathcal{B}$, with $\mathcal{B}$ locally small, has a left adjoint precisely when it preserves small limits. To prove sufficiency, start with $b:B\to UA$ and intersect all subobjects $M\hookrightarrow A$ through which $b$ factors after applying $U$. Well-poweredness makes this a small intersection of subobjects, and preservation of limits gives a smallest supporting object $(M,b_M)$. For maps $s,t:M\rightrightarrows Q_i$, equality $U(s)b_M=U(t)b_M$ makes their equalizer another supporting subobject. Minimality forces that equalizer to be invertible, so $s=t$. Thereforeis an injective function. The evaluation embedding into cogenerator products embeds $M$ in ${\prod }_i{Q}_i^{S_i}$, where $S_i$ is the realized subset of the fixed set $\mathcal{B}(B,U{Q}_i)$. There are only a set of choices of these subsets, only a set of subobjects of each resulting product, and only a set of arrows from $B$ to each $U$-image. They give a solution set, so the already proved general adjoint functor theorem applies. Necessity is again preservation of limits by a right adjoint. For $\mathbf{KHaus}$, monomorphisms are embeddings onto closed subspaces, and $[0,1]$ is a coseparator because continuous functions to it separate points. Thus the Special adjoint functor theorem gives the same reflection directly.

For the covariant category in this part, put $\mathcal{E}=[\mathcal{C},\mathbf{Set}]$ and $r_c=\mathcal{C}(c,-)$. Its global-sections functor is $\Gamma =\mathrm{Hom}(1,-)$, the limit of the diagram. If $\Gamma$ is also an inverse image, it is a left adjoint and preserves all small colimits.

Consider the canonical pointwise-surjective mapAt $d$, the summand $r_d(d)$ contains $1_d$, so $q$ is an epimorphism. In a presheaf topos epimorphisms are regular, and a colimit-preserving functor preserves their coequalizer presentations. Thus $\Gamma q$ is surjective. Since $\Gamma$ preserves coproducts, some summand has a global element, yielding a section $s:1\to r_c$ of its unique map $r_c\to 1$. The constant singleton functor is a retract of a covariant representable.

Conversely, if $1$ is such a retract in a category, $\mathrm{Hom}(1,-)$ is a retract of $\mathrm{Hom}(r_c,-)$, which is evaluation at $c$ by the Yoneda lemma. Evaluation preserves colimits pointwise. The colimit comparison for a retract functor is a retract of the evaluation comparison; a retract of an isomorphism is an isomorphism, so $\Gamma$ preserves all small colimits. It also preserves finite limits because it is representable. A right adjoint can be constructed byFor $h:d\to d^{\prime }$, precomposition gives $r_{d^{\prime }}\to r_d$, and applying $\Gamma$ and then $\mathrm{Set}(-,S)$ gives $R(S)(d)\to R(S)(d^{\prime })$. The presentation of every functor as a colimit of representables gives $\mathrm{Hom}(F,R(S))\cong \mathrm{Set}(\Gamma F,S)$. Hence $\Gamma ⊣R$, so $\Gamma$ is an inverse image and the topos is local.

The retract condition has a precise interpretation as the initial object criterion for a covariant local topos. Write the section as a natural family $u_d:c\to d$. Naturality says $h{u}_d=u_{d^{\prime }}$ for every $h:d\to d^{\prime }$. Put $e=u_c$. Then $e^2=e$ and every $f:c\to d$ satisfies $fe=u_d$. In the idempotent completion of $\mathcal{C}$, the object $(c,e)$ is initial: an arrow from it to $(d,k)$ must satisfy $v=kve$, and the identities force $v=u_d$; this arrow exists because $k{u}_d=u_d$ and $u_{d}e=u_d$.

Conversely an initial object $(c,e)$ in the idempotent completion gives that natural family and the retract. ThereforeIf $\mathcal{C}$ is already idempotent-complete, this is equivalent to an initial object in $\mathcal{C}$ itself, and $\Gamma$ is evaluation there. The word is initial, not terminal, because this question uses covariant functors.

One cannot omit idempotent completion for a general small category. Take the one-object category of the monoid $\{1,0\}$ with absorbing zero. It has no initial object because its endomorphism set has two elements, but the natural family $u=0$ satisfies $m0=0$ for every $m$. Its covariant functor category is local. Concretely, a functor is a set with an idempotent operator, and its global sections are the fixed points of that operator.

For a topological space $X$, there is an equally explicit criterion. If $\mathbf{Sh}(X)$ is local, apply $\Gamma$ to the epimorphism ${\coprod }_i{U}_i\to 1$ associated with any open cover $X={\bigcup }_i{U}_i$, regarding the opens as subterminal sheaves. Since $\Gamma U_i$ is a singleton exactly when $U_i=X$, and otherwise empty, coproduct and epimorphism preservation force some $U_i=X$.

Thus every open cover of $X$ contains $X$ itself. Conversely this open cover criterion for a local sheaf topos condition is equivalent to existence of a point $x$ whose only open neighborhood is $X$: the union of all proper open sets cannot be all of $X$, so choose $x$ outside it. Such a point plainly forces any cover to contain $X$. Empty spaces fail the condition.

At this point, the stalk functor $F\mapsto F_x$ is simply $F(X)=\Gamma F$, since there is only one open neighborhood. Its right adjoint is the skyscraper sheaf of setsA morphism $F\to R(S)$ is determined exactly by its map $F(X)\to S$, because all proper opens omit $x$. Hence $\Gamma ⊣R$, and $\Gamma$ preserves finite limits, proving locality. ConsequentlyIn a $T_0$ space this distinguished point is unique and closed. In a $T_1$ space the condition forces $X$ to be a singleton. Indistinguishable points in a non-$T_0$ space can all have this property; uniqueness of an actual point is not part of the general criterion.

A geometric formula is built from atomic formulas, including equality, using finite conjunctions, arbitrary set-indexed disjunctions and existential quantification in a finite variable context. Truth is the empty conjunction and falsity the empty disjunction. A geometric theory is a set of sequents $\varphi {⊢}_{\vec{x}}\psi$ between such formulas over a many-sorted signature. The context supplies the universal force of an axiom; unrestricted universal quantification, implication and negation are not formula constructors in this fragment.

These choices are exactly suited to inverse image functors of geometric morphisms. Finite-limit preservation handles equality and conjunction; colimit and image preservation handle disjunction and existential quantification. Therefore an inverse image carries an internal model of a geometric theory to another model.

Construct the geometric syntactic category ${\mathcal{C}}_{\mathbb{T}}$ as follows. Objects are formulas in context $\{\vec{x}.\varphi \}$, modulo provable renaming and equivalence. A morphism to $\{\vec{y}.\psi \}$ is an equivalence class of formulas $\theta (\vec{x},\vec{y})$ that are provably total and single-valued:The last notation abbreviates componentwise equality. Identity is equality of the context variables; composition is existential conjunction over the intermediate tuple. The category has finite limits, formed through conjunction and equality.

Give it the geometric syntactic topology $J_{\mathbb{T}}$. A family of arrows represented by ${\theta }_i({\vec{y}}_i,\vec{x})$ into $\{\vec{x}.\varphi \}$ covers whenPullback stability is substitution, and transitivity comes from distributing existential conjunction through the covering disjunctions. Thus the generated sieves define a Grothendieck topology. Empty covers impose the interpretation of falsity as the initial object.

The classifying topos isIts universal model $U_{\mathbb{T}}$ assigns a sort $A$ the sheafified representable of $\{x^A.\top \}$, functions their definable graph morphisms and relations their definable subobjects. The syntactic covering conditions make the axioms valid in this model.

For the universal property, interpreting formulas in a model $M$ in a Grothendieck topos $\mathcal{F}$ gives a finite-limit-preserving, $J_{\mathbb{T}}$-continuous functor ${\mathcal{C}}_{\mathbb{T}}\to \mathcal{F}$: covers go to jointly epimorphic families precisely because the relevant sequents hold. The Diaconescu equivalence for geometric morphisms associates to this functor a geometric morphism $\mathcal{F}\to \mathbf{Set}[\mathbb{T}]$. Conversely, pulling $U_{\mathbb{T}}$ back along a geometric morphism gives a model. The constructions on models and their homomorphisms are inverse up to natural isomorphism, yieldingnaturally in $\mathcal{F}$. This proves the required classifying property, not merely classification of set-valued models.

The limit form of the Special adjoint functor theorem is as follows. Let $\mathcal{C}$ be a locally small category, a complete category, and a well-powered category, and suppose it has a small cogenerating family $(Q_i{)}_{i\in I}$. Let $\mathcal{D}$ be a locally small category. Then a functor $G:\mathcal{C}\to \mathcal{D}$ has a left adjoint if and only if it preserves all small categorical limits. The dual exchanges completeness, well-poweredness and cogenerators for cocompleteness, well-copoweredness and generators, and characterizes functors with a right adjoint.

Here a cogenerating set distinguishes unequal parallel maps by postcomposition into one of the $Q_i$. A well-powered category has a set of subobjects of each fixed object. Completeness means existence of all small categorical limits. We use the initial-object lemma for complete categories with a weakly initial set: a complete category that is locally small has an initial object exactly when it has a weakly initial set. A proof is given in Question 5(c). We also use the fact that when $G$ preserves small categorical limits, the comma category $(B\downarrow G)$ is complete, with its limits constructed in $\mathcal{C}$; local smallness is inherited from $\mathcal{C}$. Finally, having an initial object in $(B\downarrow G)$ for each $B$ is the universal arrow from an object to a functor characterization of a left adjoint.

Assume $G$ preserves small categorical limits, and fix $B\in \mathcal{D}$. We will construct a weakly initial set in $(B\downarrow G)$, rather than assume a solution set. Start with an arbitrary object $(C,x)$, where $x:B\to GC$. Call a subobject $m:S\hookrightarrow C$ supporting if $x=G(m)x_S$ for some $x_S:B\to GS$. There is at least one, namely $1_C$. Because $G$ preserves pullbacks in a category, it preserves monomorphisms: the self-pullback characterization of a monic map is preserved. Consequently each $x_S$ is unique.

There is a set of supporting subobjects by well-poweredness. Their intersection of subobjects exists by completeness; write it $m_0:C_0\hookrightarrow C$. Applying $G$ to this intersection categorical limit gives a unique $x_0:B\to G{C}_0$ lifting all the $x_S$, and hence $x=G(m_0)x_0$. Minimality says that any subobject $n:R\hookrightarrow C_0$ supporting $x_0$ is invertible: $m_{0}n$ supports $x$, so $m_0$ factors through $m_{0}n$, giving a right inverse to the monic $n$.

If $u,v:C_0\to Q_i$ satisfy $G(u)x_0=G(v)x_0$, their equalizer supports $x_0$, since $G$ preserves that equalizer. It is therefore an isomorphism, and $u=v$. We have obtained an injective functionLet $J_i$ be its image, a subset of the fixed set $S_i=\mathcal{D}(B,G{Q}_i)$. The evaluation embedding into cogenerator products is a monomorphismIndeed each $s\in J_i$ corresponds to a unique map $C_0\to Q_i$, and the cogenerating set property makes the resulting family jointly monic. Notice that we use the realized subsets $J_i$, not all of $S_i$: there need not be a map $C_0\to Q_i$ available for an unused index.

There is a set of tuples $J=(J_i)$, since $I$ and every $S_i$ are sets. For each resulting $P_J$, choose representatives of its subobjects; these form a set by well-poweredness. For every representative $R$, include every map $y:B\to GR$, a set by local smallness of $\mathcal{D}$. All resulting pairs $(R,y)$ form a set. Our given $(C,x)$ receives a map from such a pair: transport $x_0$ along the isomorphism between $C_0$ and the chosen representative, then compose its inverse with $m_0$. Thus these pairs form a weakly initial set in $(B\downarrow G)$. This is the cogenerator bound for comma-category solution sets.

The initial-object lemma for complete categories with a weakly initial set gives an initial object $(LB,{\eta }_B)$. For $h:B\to B^{\prime }$, define $Lh$ by the unique equationUniqueness proves the functor laws and gives the natural bijectionsThus $L⊣G$. Conversely, a right adjoint preserves small categorical limits: its adjunction identifies maps into the proposed limiting object with compatible families of maps into the diagram. This proves both directions of the limit form of the special adjoint functor theorem. Applying the argument to the opposite categories proves the dual statement. Choices of representatives and adjoint objects are understood in the usual ambient-universe convention for large categories.

Suppose $G$ is a representable functor, with a natural isomorphism $\theta :\mathcal{C}(R,-)\to G$. Put $r={\theta }_R(1_R)$. For any $(A,a)$ in the comma category $(1\downarrow G)$, the representing bijection gives a unique $f:R\to A$ with ${\theta }_A(f)=a$. By naturality, ${\theta }_A(f)=G(f)(r)$, exactly the condition for $f:(R,r)\to (A,a)$. Hence $(R,r)$ is an initial object.

Conversely, if $(R,r)$ is an initial object, thenis a bijection for each $A$: existence and uniqueness are precisely initiality applied to every $(A,a)$. For $h:A\to B$, $G(h){\theta }_A(f)=G(hf)(r)={\theta }_B(hf)$, proving naturality. Thus $r$ is a universal element and $\theta$ is a representation of a functor. Being representable is equivalent to the singleton comma category having an initial object.

A weakly initial set is a set of objects $(W_i{)}_{i\in I}$ such that every object receives at least one map from some $W_i$. An initial object is itself a singleton weakly initial set. We prove the converse without replacing uniqueness by weak initiality.

Let $\mathcal{E}$ be a complete category that is locally small. Form $W={\prod }_i{W}_i$. For every object $A$, choose $W_i\to A$ and compose with the projection $W\to W_i$, so $W$ is weakly initial. Completeness supplies a terminal object, so the given weakly initial set cannot be empty.

The endomorphisms of $W$ form a set. Take a simultaneous equalizer $e:I\to W$ of all of them with $1_W$; equivalently, equalize the two maps from $W$ into ${\prod }_{s\in \mathcal{E}(W,W)}W$ with components $s$ and $1_W$. Thus $se=e$ for every endomorphism $s$. The object $I$ is weakly initial, since $I\stackrel{e}{\to }W\to A$ exists for every $A$. Weak initiality of $W$ gives a map $u:W\to I$. Applying the equalizing identity to $eu$ yields $eue=e$, so monicity of $e$ gives $ue=1_I$.

For any endomorphism $v:I\to I$, the map $evu$ is an endomorphism of $W$. Therefore $evue=e$. Since $ue=1_I$, this becomes $ev=e$, giving $v=1_I$. Now let $f,g:I\rightrightarrows A$ and form their equalizer $j:E\to I$. Weak initiality of $I$ gives $h:I\to E$. The composite $jh$ is an endomorphism of $I$, hence the identity. Consequently $f=fjh=gjh=g$.

There is at least one map from $I$ to every object, and the argument proves that there is at most one. The object $I$ is an initial object. This establishes the initial-object lemma for complete categories with a weakly initial set using only small products in a category and equalizers.

If $G$ is a representable functor, it preserves small categorical limits. Part (b) makes $(1\downarrow G)$ have an initial object, which gives a weakly initial set.

Conversely, assume limit preservation and a weakly initial set in $(1\downarrow G)$. The needed comma-limit fact is explicit: for a small diagram in a category $(A_j,a_j)$, form $L={\lim }_j{A}_j$ in $\mathcal{C}$. The elements $a_j\in G(A_j)$ form a compatible family, hence determine a unique $a\in G(L)$ under $G(L)\cong {\lim }_{j}G(A_j)$. The projections from $(L,a)$ are a categorical limit in the comma category. This includes the empty diagram, since $G$ carries the terminal object to a singleton. Each comma hom-set is a subset of a hom-set in $\mathcal{C}$, so the comma category is also locally small.

Part (c) now gives an initial object of $(1\downarrow G)$, and part (b) supplies a representation of a functor. ThereforeThe size requirement is a genuine solution-set condition; limit preservation alone does not produce it in an arbitrary complete category.