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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.

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.

We prove both versions. In the general adjoint functor theorem, let $G:\mathcal{D}\to \mathcal{C}$ be a functor between locally small categories, with $\mathcal{D}$ complete. Then $G$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition: for each $C\in \mathcal{C}$ there is a set of arrows $u_i:C\to G{D}_i$ such that every arrow $u:C\to GD$ factors as $G(h)u_i$ for some $i$ and $h:D_i\to D$.

First prove the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{A}$ be locally small and complete, and have a small weakly initial family $(W_i)$. Its product $W$ is weakly initial: a map $W_i\to A$ can be composed with the product projection. Take $e:E\to W$ to be the simultaneous equalizer of every endomorphism of $W$ and its identity. This is a small limit since $\mathcal{A}(W,W)$ is a set. Maps $E\to A$ exist by composing $e$ with a weakly initial map $W\to A$.

For uniqueness, take any $a,b:E\to A$ and their equalizer $j:Z\to E$. Weak initiality of $W$ gives $t:W\to Z$. The endomorphism $ejt$ of $W$ fixes $e$ by its defining equalizer property, so $ejte=e$. Since $e$ is monic, $jte=1_E$. Thus $j$ is both monic and split epic, hence invertible, forcing $a=b$. Therefore $E$ is initial. This proves the lemma without an unproved initial-object assertion.

Now fix $C\in \mathcal{C}$. The comma category $(C\downarrow G)$ is locally small. It is complete: take the limit of the underlying diagram in $\mathcal{D}$, use preservation by $G$ to factor the compatible arrows from $C$, and obtain the universal comma cone. The solution set is exactly a small weakly initial family in this comma category. The lemma provides its initial object $(FC,{\eta }_C)$. Its universal property is the natural bijection $\mathcal{D}(FC,D)\cong \mathcal{C}(C,GD)$. For $v:C\to C^{\prime }$, initiality gives the unique $Fv$ satisfying $G(Fv){\eta }_C={\eta }_{C^{\prime }}v$; uniqueness proves identities and composition. Thus these objects form a left adjoint. Conversely a left adjoint provides the singleton solution set consisting of its unit arrow, and a right adjoint preserves limits: the adjunction bijections identify cones into the image diagram with cones from the corresponding left-adjoint object into the original diagram. This proves both directions of the general theorem.

The limit form of the special adjoint functor theorem assumes instead that $\mathcal{D}$ is complete, locally small and well-powered, and has a small cogenerating family $(Q_i)$. For any locally small target, a functor $G:\mathcal{D}\to \mathcal{C}$ has a left adjoint exactly when it preserves small limits. Necessity is already proved. For sufficiency we construct a solution set, so that the general theorem applies.

Fix $C$. Given $u:C\to GD$, consider all pairs $h,k:D\to Q_i$ with $G(h)u=G(k)u$. There is only a set of these pairs. Intersect their equalizers to obtain a monomorphism $m:D^{\prime }\to D$. Since $G$ preserves this limit and $u$ equalizes the images of all pairs, it factors uniquely as $u=G(m)u^{\prime }$ for $u^{\prime }:C\to G{D}^{\prime }$.

For each realized pair $(i,v)$ with $v=G(h)u:C\to G{Q}_i$, choose one such $h$. Any other $k$ with the same pair restricts to the same map on $D^{\prime }$. The restricted chosen maps jointly separate morphisms into $D^{\prime }$: if two such morphisms become equal after all chosen maps, they become equal after every $hm$, and the cogenerating family then makes their composites with $m$ equal; monicity of $m$ cancels it. Hence the resulting mapis monic, where $J$ is a subset of the fixed set ${\coprod }_i\mathcal{C}(C,G{Q}_i)$.

There is only a set of possible $J$, a set of subobjects of each corresponding product by well-poweredness, and a set of arrows from $C$ into $G$ of each chosen subobject representative by local smallness of $\mathcal{C}$. Collect all these representative comma objects. The factorization just constructed shows that they form a weakly initial set in $(C\downarrow G)$. This is the cogenerator bound for comma-category solution sets, with realized subsets avoiding any assumption that all proposed maps to cogenerators have lifts. The general theorem now gives the left adjoint and proves the special theorem.

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.