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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.

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.

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

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

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

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

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

A monomorphism $m:A\to B$ is left cancellable: $mu=mv$ implies $u=v$ for all parallel arrows into $A$. A strong monomorphism is a monomorphism with the following lifting property: if $e:X\to Y$ is an epimorphism and $u:X\to A$, $v:Y\to B$ satisfy $mu=ve$, there is a diagonal $d:Y\to A$ with $de=u$ and $md=v$. It is unique by monicity. An extremal monomorphism is a monomorphism $m$ for which every factorization $m=he$ with $e$ an epimorphism forces $e$ to be an isomorphism. We use the strong lifting formulation for the paper's terminology; below we also justify its equivalence with the extremal formulation under the complete, well-powered hypotheses. A regular monomorphism is an equalizer of two parallel arrows.

First, a strong monomorphism is extremal. In a factorization $m=he$, apply its lifting property with upper arrow $1_A$ and lower arrow $h$. This gives $d$ with $de=1_A$. Since $e$ is an epimorphism, $ede=e$ implies $ed=1$, so $e$ is an isomorphism.

If $m:A\to B$ is an equalizer of $r,s:B\rightrightarrows C$, consider a lifting square as above. The equation $rve=smu=sve$ and the epimorphism property give $rv=sv$. The equalizer property therefore gives a unique $d$ with $md=v$. Monicity then gives $de=u$. Hence every regular monomorphism is a strong monomorphism.

For the intersection of strong subobjects, let $m_i:A_i\to B$ be strong and let $p_i:P\to A_i$ exhibit their intersection, with $m=m_1{p}_1=m_2{p}_2$. Given $mu=ve$ with $e$ an epimorphism, the two strong monomorphisms supply diagonals $d_i:Y\to A_i$ satisfying $d_{i}e=p_{i}u$ and $m_i{d}_i=v$. The pullback in a category property supplies $d:Y\to P$ with $p_{i}d=d_i$. Its two projections show $de=u$, and $md=v$. Thus the intersection is strong. The same argument works for every existing small intersection of strong monomorphisms. Also, composites of strong monomorphisms are strong: first lift through the outer one, then through the inner one.

Write $\mathcal{S}$ for the full subcategory formed by every saturated object with respect to anodyne morphisms, using the paper's anodyne morphism in a category convention. Let $m:C\to B$ be a strong monomorphism and $B$ saturated. For an anodyne morphism in a category $e:X\to Y$ and $u:X\to C$, saturation extends $mu$ to $v:Y\to B$. Since $e$ is an epimorphism, the strong lifting property supplies $d:Y\to C$ extending $u$. Therefore every strong subobject of a saturated object is saturated.

Now assume $\mathcal{C}$ is a complete category and a well-powered category, with the given saturated embeddings. Choose a monomorphism $a:A\to B$ into a saturated object. Intersect all strong subobjects of $B$ through which $a$ factors. They form a set by the well-powered category hypothesis, and the identity of $B$ is one of them. Completeness constructs their intersection, so we obtainBy the intersection of strong subobjects, $m$ is strong; hence $LA$ is saturated. Since $a$ is a monomorphism, so is ${\eta }_A$.

To prove that ${\eta }_A$ is an epimorphism, take $u,v:LA\rightrightarrows D$ with $u{\eta }_A=v{\eta }_A$. Their equalizer $j:E\to LA$ is a regular monomorphism, and ${\eta }_A$ factors through it. The composite $mj:E\to B$ is strong, so it belongs to the family defining $LA$. Minimality gives $t:LA\to E$ with $mjt=m$. Cancelling $m$ gives $jt=1_{LA}$; since $j$ is a monomorphism, also $tj=1_E$. Thus $j$ is an isomorphism and $u=v$. We have provedThis is the saturated reflection from a strong-subobject intersection.

The construction of a least strong subobject and the equalizer argument work for any morphism into any object, without requiring the source map to be monic or the target to be saturated. They give an epimorphism followed by a strong monomorphism. Applying this factorization to an extremal monomorphism makes the first factor invertible; thus the extremal and strong formulations agree under the present complete, well-powered hypotheses.

For every saturated $S$, saturation extends every $f:A\to S$ across the anodyne morphism in a category ${\eta }_A$ to $\bar{f}:LA\to S$. Its extension is unique because ${\eta }_A$ is an epimorphism. Consequentlynaturally in $S$ and $A$. In particular, for $h:A\to A^{\prime }$ define $Lh$ by $Lh{\eta }_A={\eta }_{A^{\prime }}h$; uniqueness proves the identity and composition laws. This constructs the reflector and proves that $\mathcal{S}$ is a reflective subcategory.

Finally we prove that $\mathcal{S}$ is a balanced category. If $e:U\to V$ is epic in $\mathcal{S}$ and $u,v:V\to X$ satisfy $ue=ve$, embed $X$ by a monomorphism $j:X\to Y$ into a saturated object. Fullness makes $ju,jv$ arrows of $\mathcal{S}$, so $ju=jv$ and hence $u=v$. Thus an epimorphism in $\mathcal{S}$ is also epic in $\mathcal{C}$.

A monomorphism $f:U\to V$ in $\mathcal{S}$ is also monic in $\mathcal{C}$: for $a,b:X\to U$, extend them uniquely along ${\eta }_X$ to $\bar{a},\bar{b}:LX\to U$. If $fa=fb$, cancellation of the ambient epimorphism ${\eta }_X$ gives $f\bar{a}=f\bar{b}$, so monicity in $\mathcal{S}$ gives $\bar{a}=\bar{b}$ and $a=b$. Now if $f$ is both monic and epic in $\mathcal{S}$, it is an anodyne morphism in a category in $\mathcal{C}$. Saturation of $U$ extends $1_U$ across $f$ to $r:V\to U$ with $rf=1_U$. Ambient epicity gives $fr=1_V$. Fullness puts this inverse in $\mathcal{S}$, proving

On a complete category that is locally small, a set-valued functor is representable exactly when it preserves small categorical limits and its category of elements has a weakly initial set. Categorical limit preservation makes that comma category complete. The initial-object lemma for complete categories with a weakly initial set then supplies a universal element, hence a representation of a functor.