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Under the hypotheses of the limit form of the special adjoint functor theorem, intersect all subobjects of $C$ supporting $x:B\to GC$. Limit preservation gives a smallest supporting $(C_0,x_0)$. If $G(u)x_0=G(v)x_0$ for maps $u,v:C_0\to Q_i$, their equalizer supports $x_0$, so minimality makes it invertible and $u=v$. Thus the indicated map of hom-sets is injective. The evaluation embedding into cogenerator products embeds $C_0$ in a product of cogenerators indexed by the realized subsets of the fixed sets $\mathcal{D}(B,G{Q}_i)$. There are only a set of such products, a set of their subobjects, and a set of maps from $B$ into their images under $G$. These data give a weakly initial set in $(B\downarrow G)$. Using realized subsets avoids assuming maps into every cogenerator exist.

A locally small category with all small categorical limits and a small weakly initial set has an initial object. Take the product $W$ of the weakly initial family, then the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ and its identity. For any parallel $a,b:E\to X$, their equalizer $j:Y\to E$ receives a map $t:W\to Y$. The equation $(ejt)e=e$ forces $jte=1_E$, so $j$ is invertible and $a=b$. Weak initiality supplies existence of maps from $E$. This is the smallness mechanism in the Freyd general adjoint functor theorem.

A functor $G:\mathcal{C}\to \mathcal{D}$ from a locally small category that is complete and well-powered, with a small cogenerating family, to a locally small category has a left adjoint exactly when it preserves small categorical limits. For sufficiency, the cogenerator bound for comma-category solution sets produces a weakly initial set in each complete comma category $(B\downarrow G)$. The initial-object lemma for complete categories with a weakly initial set supplies its initial object, a universal arrow from an object to a functor. Necessity is limit preservation by a right adjoint.

Only $L\Rightarrow A$ remains. Assume $U$ preserves small categorical limits, and choose a small cogenerating family $(Q_i{)}_{i\in I}$ in $\mathcal{C}$. Fix a set $X$. The comma category $(X\downarrow U)$ is a complete category: underlying limits in $\mathcal{C}$, together with limit preservation by $U$, lift compatible maps from $X$. It is locally small, since its arrows form subsets of the hom-sets in $\mathcal{C}$.

We construct a weakly initial set in this comma category. Given $f:X\to U(C)$, intersect all subobjects $m:B\hookrightarrow C$ through which $f$ lifts under $U$. There are only a set of subobjects because $\mathcal{C}$ is well-powered; the family includes $1_C$. Their intersection exists by completeness. Also $U$ preserves monomorphisms, since a map is monic precisely when its self-pullback diagonal is invertible, and $U$ preserves such pullbacks.

Thus all lifts of $f$ are unique and compatible, and preservation of the intersection givesNo proper subobject of $C_0$ supports a lift of $f_0$: its composite into $C$ would belong to the intersected family, forcing it to contain $C_0$. This is a minimal supported subobject.

For each $i$, the mapis injective. Indeed, equal images mean $f_0$ lifts to the equalizer of the two maps, because $U$ preserves that equalizer. Minimality forces the equalizer to be invertible, so the maps agree.

The cogenerator property makes the evaluation mapa monomorphism: any two arrows into $C_0$ that agree after all these projections are equal. The indexing pairs $(i,h)$ inject into the fixed setTheir image is some subset $J\subseteq B_X$. Consequently $C_0$ is isomorphic to a subobject of a product $P_J$ of the $Q_i$ indexed by $J$. There are only a set of such subsets $J$, only a set of subobjects of each $P_J$, and only a set of maps $X\to U(B)$ for each representative subobject $B$.

Collect all these pairs $(B,x:X\to U(B))$. Every $(C,f)$ receives a comma arrow from one of them, via the isomorphic copy of $(C_0,f_0)$ and $m_0$. Hence they form a weakly initial set. The initial-object lemma for complete categories with a weakly initial set, proved explicitly in Question 8, gives an initial object of $(X\downarrow U)$. Its map from $X$ is a universal arrow from an object to a functor. Choosing these universal arrows for all $X$ defines a left adjoint to $U$.

ThusThe products indexed by realized subsets $J$ are important: one cannot assume that $C_0$ has a map to every cogenerator and simply add arbitrary missing coordinates.

The general adjoint functor theorem says that, for a functor $U:\mathcal{C}\to \mathcal{D}$ with $\mathcal{C}$ a complete category and both categories locally small, $U$ has a left adjoint exactly when it preserves small categorical limits and satisfies the solution-set condition. The latter means that for each $D\in \mathcal{D}$ there is a set of arrows $f_i:D\to U(C_i)$ such that every $f:D\to U(C)$ factors asEquivalently, $(D\downarrow U)$ has a weakly initial set.

The limit form of the special adjoint functor theorem says that, if in addition $\mathcal{C}$ is well-powered and has a small cogenerating family, then $U$ has a left adjoint exactly when it preserves small limits. These are the limit/left-adjoint forms; their duals interchange limits with colimits, left with right adjoints, and cogenerators with generators.

For the first route, let $\mathcal{E}$ be a complete category that is locally small and has a weakly initial set $(W_i)$. Put $P={\prod }_i{W}_i$. It is weakly initial: for any $X$, choose an arrow $W_i\to X$ and precompose with the corresponding product projection. The set $\mathcal{E}(P,P)$ indexes a simultaneous equalizerIt exists by completeness, for example as the equalizer of the two maps $P\rightrightarrows {\prod }_{u:P\to P}P$ with components $u$ and $1_P$. Since $E\to P\to X$ exists for each $X$, $E$ is also weakly initial.

Choose $r:P\to E$. Applying the defining equation to $er:P\to P$ gives $ere=e$, whence $re=1_E$ because $e$ is a monomorphism. If $v:E\to E$ is any endomorphism, apply it to $evr:P\to P$ to obtain $evre=e$, hence $ev=e$ and $v=1_E$. Thus $E$ has only its identity endomorphism.

For two arrows $a,b:E\rightrightarrows X$, take their equalizer $j:Y\to E$. Weak initiality of $E$ gives $t:E\to Y$. Since $jt$ is an endomorphism of $E$, $jt=1_E$. The monic arrow $j$ with this right inverse is invertible; therefore $a=b$. Existence of an arrow $E\to X$ and this uniqueness show that $E$ is initial.

For the general adjoint functor theorem, limit preservation makes each comma category $(D\downarrow U)$ complete, and the solution-set condition makes it weakly initially generated by a set. The lemma provides an initial object $(FD,{\eta }_D)$. For $v:D\to D^{\prime }$, the universal property uniquely defines $Fv:FD\to F{D}^{\prime }$ byUniqueness gives the functor laws. The same property gives natural bijectionsso $F⊣U$. Conversely a right adjoint preserves limits, by applying its hom-set bijections to cones, and its unit $(FD,{\eta }_D)$ is already an initial object of each comma category, hence a singleton solution set. This proves necessity as well as sufficiency.

For the second route, assume the general adjoint functor theorem and the additional hypotheses of the Special adjoint functor theorem. Let $(Q_i{)}_{i\in I}$ be a small cogenerating family. Fix $D\in \mathcal{D}$ and an arrow $f:D\to UC$. A limit-preserving functor preserves monomorphisms, by the self-pullback criterion. Intersect the set of all subobjects $B\hookrightarrow C$ supporting $f$. Preservation of this categorical limit gives a minimal supporting pair $(C_0,f_0:D\to U{C}_0)$ and a monomorphism $m_0:C_0\to C$ with $U(m_0)f_0=f$.

If $g,h:C_0\rightrightarrows Q_i$ have $Ug{f}_0=Uh{f}_0$, their equalizer supports $f_0$ and must be invertible by minimality. ThusThe evaluation map into the product of all these cogenerator targets is monic, because the family coseparates arrows. Its coordinates inject into the fixed setFor every subset $J\subseteq B_D$, form the product $P_J$ whose coordinate $(i,z)$ has target $Q_i$. Each minimal $C_0$ is isomorphic to a subobject of one such product, using precisely its realized coordinates. There is a set of subsets $J$; the well-powered category condition gives a set of representative subobjects of each product. For each representative $B$, include every arrow $D\to UB$, a set by local smallness of $\mathcal{D}$.

This is a solution-set condition: the minimal pair is isomorphic to one of these representatives, and its inclusion into $C$ supplies the required factorization of $f$. The general adjoint functor theorem now gives the left adjoint. This is the cogenerator bound for comma-category solution sets proof of the special theorem. Necessity follows again because a right adjoint preserves limits. Both alternative proof routes are therefore supplied.

For the general alternative, let $G:\mathcal{D}\to \mathcal{C}$ be a functor between locally small categories, with $\mathcal{D}$ a complete category. The Freyd general adjoint functor theorem says that $G$ has a left adjoint if and only if it preserves small categorical limits and satisfies the solution-set condition. Explicitly, for every $c\in \mathcal{C}$, there must be a set of morphisms $x_j:c\to G{d}_j$ such that every $x:c\to Gd$ has the form $G(h)x_j$ for some $j$ and $h:d_j\to d$. Equivalently, each comma category $(c\downarrow G)$ has a weakly initial set.

We prove the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{E}$ be a locally small category which is complete and has a weakly initial set $(w_j{)}_{j\in J}$. Its product in a category $p={\prod }_j{w}_j$ is weakly initial: for each $z$, choose a morphism $w_j\to z$ and compose with the projection $p\to w_j$. The hom-set $\mathcal{E}(p,p)$ is a set. Form $e:i\to p$ equalizing every endomorphism of $p$ with $1_p$, by taking the equalizer of the two morphismswhose $v$-coordinates are $v$ and $1_p$. Then $ve=e$ for every endomorphism $v$, and $e$ is monic. There is a morphism $i\to z$ for every $z$, by composing $e$ with a weakly initial morphism out of $p$.

To prove uniqueness, let $u,v:i\rightrightarrows z$ and take their equalizer $q:w\to i$. Weak initiality of $p$ gives $a:p\to w$. The composite $eqa:p\to p$ is an endomorphism, so $eqae=e$. Cancel the monomorphism $e$ to obtain $qae=1_i$. Thus $q$ is both monic and split epic, hence an isomorphism: if $qk=1$, cancellation from $qkq=q$ gives $kq=1$. Since $uq=vq$, it follows that $u=v$. Therefore $i$ is an initial object, proving the lemma fully.

Now assume categorical limit preservation and the solution-set condition. The comma category $(c\downarrow G)$ is locally small. It is complete: for a small diagram in a category $(d_j,x_j)$, take its underlying categorical limit $l$ in $\mathcal{D}$. Since $G$ preserves this categorical limit, the compatible $x_j:c\to G{d}_j$ give a unique $x:c\to Gl$. The same universal property shows that $(l,x)$ is the categorical limit in the comma category. Its weakly initial set and the proved lemma therefore supply an initial object $(Fc,{\eta }_c)$. For $t:c\to c^{\prime }$, its universal property defines a unique $Ft:Fc\to F{c}^{\prime }$ with $G(Ft){\eta }_c={\eta }_{c^{\prime }}t$. Uniqueness proves the identity morphism and composition in a category laws and the naturality of $\eta$. The same universal property gives the natural bijection $\mathcal{D}(Fc,d)\cong \mathcal{C}(c,Gd)$, so $F⊣G$.

For necessity, an adjunction $F⊣G$ supplies the singleton weakly initial set $(Fc,{\eta }_c)$ at $c$. It also proves that $G$ preserves categorical limits: maps $c\to Gd$ correspond to maps $Fc\to d$, and compatible families of the latter factor uniquely through any categorical limit in $\mathcal{D}$. Hence the image categorical cone under $G$ has the required universal property. This proves both directions of the general alternative.

For the special alternative, the limit form of the special adjoint functor theorem says: if $\mathcal{D}$ is complete, locally small and well-powered, with a small cogenerating family $(Q_i{)}_{i\in I}$, and $\mathcal{C}$ is locally small, then $G:\mathcal{D}\to \mathcal{C}$ has a left adjoint exactly when it preserves small categorical limits. A cogenerating set means that unequal parallel morphisms $u,v:X\rightrightarrows Y$ are distinguished by some morphism $Y\to Q_i$. We prove the needed solution-set condition, so that the general theorem applies.

Fix $x:c\to Gd$. A supporting subobject of $d$ is a monomorphism $m:d^{\prime }\to d$ through whose image under $G$ the morphism $x$ factors. There is at least one, namely $1_d$. Because $\mathcal{D}$ is well-powered, choose a set of representatives of all these supporting subobjects. A limit-preserving functor preserves monomorphisms: a map is monic exactly when its diagonal into its self-pullback in a category is an isomorphism, and both the pullback in a category and this isomorphism are preserved. Consequently each factorization of $x$ through $Gm$ is unique. Form the intersection of subobjects of the supporting representatives, as a wide pullback in a category over $d$. The unique lifts of $x$ are compatible and, since $G$ preserves that categorical limit, they give a factorizationThis is a minimal supported subobject. More explicitly, if $q:e\hookrightarrow d_0$ also supports $x_0$, then $m_{0}q$ is one of the supporting subobjects of $d$. The intersection therefore factors through $m_{0}q$, giving $r:d_0\to e$ with $m_{0}qr=m_0$. Monicity of $m_0$ gives $qr=1_{d_0}$; monicity of $q$ then makes $q$ invertible.

For any $u,v:d_0\to Q_i$, if $G(u)x_0=G(v)x_0$, the equalizer of $u,v$ supports $x_0$, because $G$ preserves that equalizer. Minimality makes it invertible, so $u=v$. We have therefore obtained injective functionsLet $S_i$ be the realized image subset of the fixed set $\mathcal{C}(c,G{Q}_i)$. Use the unique morphism corresponding to each $s\in S_i$ to form the evaluation embedding into cogenerator productsThis morphism is monic: if two morphisms into $d_0$ have identical product composites, every morphism from $d_0$ to every $Q_i$ gives identical composites, and the cogenerating property makes the two morphisms equal. There is only a set of possible families of subsets $(S_i)$, and hence a set of possible products $P_S$. Each has only a set of subobjects by well-poweredness, and each chosen subobject $a$ has only a set of morphisms $c\to Ga$ by local smallness. Take all these pairs $(a,z:c\to Ga)$ as a weakly initial set. Every original $(d,x)$ receives a morphism from a pair representing its $(d_0,x_0)$, through $m_0$. Thus the set is weakly initial in $(c\downarrow G)$.

The general theorem now supplies a left adjoint. Necessity again follows from preservation of categorical limits by a right adjoint. This proves the special alternative, including its size argument. Using only the realized subsets $S_i$ avoids assuming that an arbitrary object admits a morphism into every cogenerator. Both adjoint functor theorem alternatives are proved.

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 $\mathrm{Nat}({\mathrm{Lan}}_{F}Q,G)\cong \mathrm{Nat}(Q,GF)$.

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.

The Freyd general adjoint functor theorem is as follows. Let $G:\mathcal{D}\to \mathcal{C}$, with $\mathcal{D}$ a complete category which is locally small. Then $G$ has a left adjoint if and only if it preserves all small categorical limits and satisfies the solution-set condition: for each $c\in \mathcal{C}$, there is a set-indexed familysuch that every $u:c\to Gd$ can be written $u=G(h)u_i$ for some $i$ and some $h:d_i\to d$. The target category need not be complete. We prove all the existence steps.

First establish 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 $\{w_i\}$, meaning every object receives a map from at least one $w_i$. Its categorical product $w={\prod }_i{w}_i$ is weakly initial: compose a categorical product projection with the map from the appropriate $w_i$. Local smallness makes $\mathcal{E}(w,w)$ a set. Let $e:v\to w$ be the simultaneous equaliser of every endomorphism of $w$ and the identity. Explicitly it is the equaliser of the two arrowswith components $k$ and $1_w$. Thus $ke=e$ for every endomorphism $k$, and $e$ is monic.

Every object $x$ receives a morphism from $v$, by composing $e$ with a map $w\to x$. To prove uniqueness, take parallel maps $a,b:v\to x$ and their equaliser $m:y\to v$. Weak initiality of $w$ gives $t:w\to y$. The composite $emt$ is an endomorphism of $w$, so $(emt)e=e$. Cancelling the monomorphism $e$ gives $mte=1_v$. Hence $m$ is both monic and a split epimorphism, and is therefore invertible. Since $am=bm$, this forces $a=b$. We have proved $v$ is initial, without using a class-sized categorical product or equaliser.

For sufficiency in the theorem, fix $c$ and consider the comma category $(c\downarrow G)$. Its objects are pairs $(d,u:c\to Gd)$; a morphism $h:(d,u)\to (d^{\prime },u^{\prime })$ satisfies $G(h)u=u^{\prime }$. This category is locally small because each hom-set is a subset of $\mathcal{D}(d,d^{\prime })$. It is complete: for a small diagram of such pairs, take the underlying categorical limit $d$ in $\mathcal{D}$. The arrows from $c$ form a cone to its image under $G$, and preservation of the categorical limit gives a unique $u:c\to Gd$ inducing them. The same universal property gives all mediators in the comma category. The solution-set family is a weakly initial set in this comma category, so the lemma supplies an initial pairFor $f:c\to c^{\prime }$, initiality defines a unique $Ff:Fc\to F{c}^{\prime }$ satisfying $G(Ff){\eta }_c={\eta }_{c^{\prime }}f$. Uniqueness proves the identity and composition laws, making $F$ a functor. Initiality also gives bijectionsThe definition of $Ff$ proves naturality in $c$, and postcomposition proves naturality in $d$. These bijections are the required adjunction $F⊣G$.

For necessity, if $F⊣G$, the single arrow ${\eta }_c:c\to GFc$ is a solution set: the transpose of any $u:c\to Gd$ gives its factorization through ${\eta }_c$. If $d$ is a limiting cone vertex of a small diagram $D_j$ in $\mathcal{D}$, then, naturally in $c$,These are the canonical cone-factorization bijections, so $Gd$ with its image projections is a categorical limit of $G{D}_j$. Thus $G$ preserves all small categorical limits, completing both directions of the theorem.

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.

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.

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.

An initial object by itself is a weakly initial set. For the converse, let $(W_i{)}_{i\in I}$ be a small weakly initial family and form its product in a category $W$. Given $X$, choose an arrow $W_i\to X$; its composite with the projection $W\to W_i$ shows that $W$ is weakly initial.

Local smallness makes $\mathrm{End}(W)$ a set, so completeness supplies the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ with $1_W$. Thus $he=e$ for every $h\in \mathrm{End}(W)$. The object $E$ is weakly initial because it maps to $W$.

For parallel arrows $a,b:E\to X$, take their equalizer $j:Y\to E$. Weak initiality of $W$ supplies $t:W\to Y$. The endomorphism $ejt$ of $W$ satisfies $(ejt)e=e$, and monicity of $e$ gives $jte=1_E$. Therefore $j$ is both a monomorphism and a split epimorphism, hence an isomorphism. Since $aj=bj$, we obtain $a=b$. There is already at least one arrow from $E$ to every $X$, so $E$ is initial. This proves the initial-object lemma for complete categories with a weakly initial set, with smallness used exactly where the endomorphisms are equalized.

If $G$ is representable, it is a limit-preserving functor, and its universal element gives an initial object of $(1\downarrow G)$, hence a weakly initial singleton.

Conversely, a limit-preserving $G$ makes $(1\downarrow G)$ a complete category. For a small diagram $(A_j,x_j)$, take $L={\lim }_j{A}_j$ in $\mathcal{C}$. Its distinguished elements form a compatible family in ${\lim }_{j}G{A}_j$. Categorical limit preservation gives a unique $x\in GL$ with $G{p}_j(x)=x_j$. The underlying categorical limit factorization of a categorical cone preserves this element, proving the comma-category universal property. For the empty diagram, this uses $G1\cong 1$. This is the construction of limits in a comma category of a limit-preserving functor.

The comma category is locally small since its arrows are subsets of the hom-sets in $\mathcal{C}$. By the previous part, its weakly initial set therefore yields an initial object. Part (b) then yields a representation of $G$. Thus representability from a solution set follows with all small categorical limits, rather than finite categorical limits alone.

For $A\in \mathcal{C}$, the comma category $(A\downarrow G)$ has objects $(D,u)$ with $D\in \mathcal{D}$ and $u:A\to GD$. A morphism $(D,u)\to (D^{\prime },u^{\prime })$ is $h:D\to D^{\prime }$ satisfying $Ghu=u^{\prime }$. Identities and composition are those of $\mathcal{D}$, and functoriality of $G$ verifies the condition under composition.

The Freyd general adjoint functor theorem states: if $\mathcal{D}$ is a locally small category with all small categorical limits, and $\mathcal{C}$ is locally small, then $G:\mathcal{D}\to \mathcal{C}$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition. The latter means that for each $A$ there is a set-indexed family $u_i:A\to G{D}_i$ such that every $u:A\to GD$ equals $Gh{u}_i$ for some $i$ and some $h:D_i\to D$.

For necessity, use the standard result that a right adjoint preserves categorical limits. If $F⊣G$, the singleton family containing the unit of an adjunction ${\eta }_A:A\to GFA$ is a solution set, since transposition gives $u=Gh{\eta }_A$ for a unique $h:FA\to D$.