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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.

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.

Use the covariant Yoneda embedding $Y(A)=\mathcal{C}(A,-)$. An arrow $f:B\to A$, viewed as an arrow $A\to B$ in the opposite category, induces precomposition $h\mapsto hf$. Identities and composition are preserved because composition in $\mathcal{C}$ is associative.

The covariant Yoneda lemma is the bijectionIts inverse sends $x\in F(A)$ to ${\alpha }_B(h)=F(h)(x)$. This is a natural transformation: for $k:B\to C$, functoriality gives $F(k){\alpha }_B(h)=F(kh)(x)={\alpha }_C(kh)$. Conversely, naturality of an arbitrary $\alpha$ at $h:A\to B$ gives ${\alpha }_B(h)=F(h){\alpha }_A(1_A)$. Evaluation at the identity morphism therefore makes the two constructions inverse. Their formulas also prove naturality in $F$ and, contravariantly, in $A$.

Taking $F=Y(B)$ identifies natural transformations $Y(A)\to Y(B)$ with $\mathcal{C}(B,A)$. Thus $Y$ is full and faithful, and reflects isomorphisms; two image objects are isomorphic exactly when the original objects are isomorphic. It also carries existing colimits in $\mathcal{C}$ to categorical limits of covariant representable functors: maps out of a colimit are exactly compatible families of maps out of its diagram. Equivalently, this embedding preserves existing categorical limits in ${\mathcal{C}}^{\mathrm{op}}$. The direction matters: the paper uses covariant representables, rather than the more usual contravariant Yoneda embedding.

For a diagram $D:\mathcal{J}\to \mathcal{C}$, a limit-preserving functor carries every limiting cone over a diagram to a limiting categorical cone. A limit-reflecting functor has the converse property: a categorical cone is limiting whenever its image is limiting. A limit-creating functor uniquely lifts every specified limiting categorical cone over the image diagram to a categorical cone over $D$, and the lift is limiting. These definitions concern diagrams of the stipulated shape; creation includes the lifting requirement, not merely reflection.

Let $(L,(p_j))$ be a categorical limit of $D$. The mapis a bijection: the right side consists exactly of compatible families of arrows from $A$, and the categorical limit's universal property gives their unique factorization through $L$. The bijection is induced by the categorical limit projections, so it proves that covariant representables preserve limits, including the empty diagram and its terminal object.

Construct pointwise limits in a functor category. For $H:\mathcal{J}\to [\mathcal{D},\mathcal{C}]$, choose at each object $d$ a categorical limit $L(d)$ of $j\mapsto H_j(d)$, with projections $p_j(d)$. For $u:d\to d^{\prime }$, the family $H_j(u)p_j(d)$ is compatible; define $L(u)$ uniquely byThe universal property gives $L(1_d)=1_{L(d)}$ and $L(vu)=L(v)L(u)$, since those equalities hold after every projection. Thus $L$ is a functor, and each $p_j:L\Rightarrow H_j$ is a natural transformation.

For any categorical cone $(M,(a_j))$, the pointwise categorical limits give unique maps $a(d):M(d)\to L(d)$. To check naturality, compose $L(u)a(d)$ and $a(d^{\prime })M(u)$ with every $p_j(d^{\prime })$; both become $H_j(u)a_j(d)=a_j(d^{\prime })M(u)$. The projections distinguish arrows into their categorical limit, so the two maps agree. Componentwise uniqueness gives uniqueness of the natural transformation $a$. This proves that $(L,p_j)$ really is the required categorical limit, rather than merely a family of objectwise candidates.

A specified categorical limit categorical cone in ${\mathcal{C}}^{\mathrm{ob}\mathcal{D}}$ fixes these objectwise vertices and projections. The displayed equation forces every arrow $L(u)$, and the argument forces every categorical cone factorization. Hence the forgetful functor uniquely lifts that categorical cone and is a limit-creating functor. The construction only takes small categorical limits in $\mathcal{C}$; it does not require $\mathcal{D}$ to be small. As usual, the functor categories are understood in a universe where their collections of transformations are meaningful.

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.