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A theorem giving sufficient conditions for a functor to have an adjoint functor. The general adjoint functor theorem uses a complete category, locally small categories, preservation of categorical limits and a solution-set condition. The Special adjoint functor theorem obtains that condition from a well-powered source and a small cogenerating family. Precise hypotheses matter in both forms.

For small $J,K$ and a functor $D:J\times K\to \mathcal{C}$ into a complete category, the two iterated categorical limits are canonically isomorphic. They have the same projections to every $D(j,k)$ and the same universal property. The result follows also from preservation of categorical limits by the limit functor.

A coseparator $S$ in a category distinguishes parallel arrows by postcomposition: if $u\ne v:X\to B$, some $t:B\to S$ satisfies $tu\ne tv$. Equivalently, the contravariant functor $\mathcal{C}(-,S)$ is faithful. In a complete category that is a locally small category, the evaluation arrow $B\to {\prod }_{t\in \mathcal{C}(B,S)}S$ is a monomorphism.

For $T:{\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{D}$, an end is a universal object $E$ with maps $e_C:E\to T(C,C)$ satisfying $T(1_C,f)e_C=T(f,1_D)e_D$ for every $f:C\to D$. Every other such family factors uniquely through $E$. For small $\mathcal{C}$ in a complete category, it is the equalizer of the two maps ${\prod }_{C}T(C,C)\rightrightarrows {\prod }_{f:C\to D}T(C,D)$.

In a complete category that is a locally small category with a coseparator $S$, suppose every monomorphism is a regular monomorphism. The evaluation embedding $B\to S^I$, $I=\mathcal{C}(B,S)$, is an equalizer of some pair into $Q$. Composing that pair with the evaluation embedding $Q\to S^J$ leaves the equalizer unchanged. Here powers mean products in a category indexed by sets, without assuming an exponential object exists.

In a complete category that is a locally small category with a coseparator $S$ and all monomorphisms regular, an object $A$ is an exponentiable object exactly when $\mathcal{C}(-\times A,S)$ is a representable functor. A representing object $E$ gives representations $E^I$ for maps into powers $S^I$. An equalizer presentation by powers of a coseparator then constructs a representation for maps into every target. The Yoneda lemma makes these representations functorial and supplies the right adjoint.

A monomorphism $m$ is extremal if every factorization $m=he$ with $e$ an epimorphism forces $e$ to be an isomorphism. A strong monomorphism has this property by its lifting property. In a complete category that is a well-powered category, intersecting all strong subobjects containing a given morphism yields an epimorphism–strong-monomorphism factorization; applying it to an extremal monomorphism proves that the extremal monomorphism is strong. The factorization definition and lifting definition should be kept distinct until the needed ambient hypotheses are supplied.

For a functor $U:\mathcal{C}\to \mathcal{D}$ between locally small categories, with $\mathcal{C}$ a complete category, $U$ has a left adjoint if and only if it preserves small categorical limits and satisfies the solution-set condition. The dual statement uses cocompleteness, small-colimit preservation and weakly terminal solution sets to characterize existence of a right adjoint.

In a complete category, the intersection of a set-indexed family of subobjects of $C$ is their wide pullback in a category over $C$. The resulting morphism into $C$ is a monomorphism, and it factors through every member. In a well-powered category, even the intersection of all subobjects satisfying a specified property is small.

If a full subcategory is reflective in a complete category, it is complete. For an ambient limit $L$ of a diagram of reflected objects, the reflection unit $L\to JKL$ is invertible by the hom-set characterization of reflected objects. Thus $KL$ is an internal categorical limit. If the subcategory is replete, $L$ itself belongs to it. The reflector need not preserve arbitrary limits.

Let $D:J\to [\mathcal{C},\mathcal{S}]$ be a small diagram in a category. Since $\mathcal{S}$ is a complete category, choose at each $C\in \mathcal{C}$ a categorical limitFor $f:C\to C^{\prime }$, the arrows $D_j(f)p_{j,C}$ form a cone over a diagram at $C^{\prime }$, because the diagram arrows $D_j\to D_k$ are natural transformations. There is therefore a unique arrow $L(f):L(C)\to L(C^{\prime })$ withThe universal property immediately gives $L(1_C)=1_{L(C)}$ and $L(gf)=L(g)L(f)$: both sides have the same composites with every limit projection. Thus $L$ is a functor, and the displayed equations make the $p_j$ natural transformations.

Given any categorical cone $q_j:X\to D_j$ in the functor category, its components induce unique $q_C:X(C)\to L(C)$. For $f:C\to C^{\prime }$, compare $L(f)q_C$ and $q_{C^{\prime }}X(f)$ after each $p_{j,C^{\prime }}$; naturality of $q_j$ makes their composites equal. Hence $q$ is a natural transformation, and uniqueness is componentwise. This proves that $L$ is a categorical limit in $[\mathcal{C},\mathcal{S}]$.

The functor category is complete, and every evaluation functor preserves limits, since evaluating the constructed limit at $C$ gives exactly the chosen limit in $\mathcal{S}$. This is the pointwise limits in a functor category construction; it also covers the empty diagram and its terminal object.

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.

A precise commutation of iterated categorical limits statement is the following. For small categories $\mathcal{I},\mathcal{J}$ and $F:\mathcal{I}\times \mathcal{J}\to \mathcal{D}$, if every column categorical limit and the resulting $\mathcal{J}$-limit exist, thenIf every row categorical limit and the resulting $\mathcal{I}$-limit also exist, thenEach isomorphism is the unique one identifying all projections to $F(i,j)$. Indeed, the preceding part proves that each iterated categorical limit has the universal property of the whole product diagram in a category. Its uniqueness gives the canonical isomorphism. Equivalently, categorical cones with vertex $X$ over the product diagram in a category are exactly compatible families, indexed by $j$, of categorical cones over $F(-,j)$; regrouping the same compatibility equations cannot change their representing object. In a complete category all the stated existence hypotheses hold for small $\mathcal{I},\mathcal{J}$. The theorem is about categorical limits with categorical limits, and makes no unconditional claim about commuting a categorical limit with a colimit.

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.

Let $J:\mathbf{CompHaus}\hookrightarrow \mathbf{Top}$ be the inclusion. Both categories are locally small categories. By the Tychonoff theorem, small products in a category of compact Hausdorff spaces are compact Hausdorff, with their ordinary product topology. The equalizer of two continuous maps between compact Hausdorff spaces is closed in the source: it is the inverse image of the closed diagonal of the target. Hence it is again compact Hausdorff. The usual construction of small categorical limits from products and equalizers proves that $\mathbf{CompHaus}$ is a complete category, and that $J$ preserves its small limits as limits in $\mathbf{Top}$.

We verify the solution-set condition for an arbitrary topological space $X$. Put $\kappa =|X|$ and $\theta =2^{2^{\kappa }}$. Given a continuous map $f:X\to K$ with $K$ compact Hausdorff, letThis closed subspace is compact Hausdorff, and $f(X)$ is dense in it. The allowed cardinal estimate therefore gives $|Y|\le \theta$. The map $f$ factors through the continuous map $X\to Y$ and the subspace inclusion $Y\hookrightarrow K$.

Choose a fixed set $S$ of cardinality $\theta$. Consider all compact Hausdorff topologies on subsets of $S$, together with all continuous maps from $X$ into the resulting spaces. There are only a set of such topologies and functions. Every $Y$ above is homeomorphic to one of these spaces after relabelling its underlying set. Consequently every $X\to JK$ factors through a member of this set of maps. This proves the solution-set condition, including $X=\varnothing$.

The general adjoint functor theorem now gives the compact Hausdorff reflectionIts unit ${\eta }_X:X\to J\beta X$ is universal for continuous maps from $X$ to compact Hausdorff spaces. For arbitrary $X$, this is a reflection, and the unit is not required to be a topological embedding.

Take the category of ordinals in reverse order ${\mathbf{Ord}}^{\mathrm{op}}$, with a unique arrow $\alpha \to \beta$ exactly when $\alpha \ge \beta$, and the unique functorBoth categories are complete categories. A small limit in the ordinal category is the supremum of the ordinals occurring in the diagram; the empty supremum is $0$, its terminal object. The functor $G$ preserves all these categorical limits, and both categories are locally small categories.

A left adjoint would select an initial object of ${\mathbf{Ord}}^{\mathrm{op}}$, meaning an ordinal at least as large as every ordinal. No such object exists. More specifically, any set of candidate ordinals has an upper bound $\gamma$, and none of its members maps to $\gamma +1$. Thus the solution-set condition fails. This example isolates the genuine smallness requirement in the general adjoint functor theorem.

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.

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.