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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.

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.

It is enough to prove the solution-set condition and then apply the general adjoint functor theorem. Fix $D\in \mathcal{D}$ and a morphism $d:D\to UC$. We will factor it through one object in a set depending only on $D$.

First construct a minimal supported subobject. Among the subobjects $m:M\hookrightarrow C$ for which $d=U(m)d_M$ for some $d_M:D\to UM$, include $1_C$ and take their intersection $m_0:C_0\hookrightarrow C$. This is a small intersection, by well-poweredness. It exists by completeness as the categorical limit of the diagram consisting of these monomorphisms into $C$. Its map to $C$ is a monomorphism: two maps with the same composite to $C$ have equal projections to every $M$, since each $m$ is monic, and are then equal by the categorical limit property. One can equivalently construct these intersections by pullbacks in a category and small products in a category, using stability of monomorphisms under pullback in a category.

Choose the factorizations $d_M$. They form a compatible categorical cone into the image diagram under $U$, all with common composite $d$ to $UC$. Preservation of small categorical limits yields $d_0:D\to U{C}_0$ with $U(m_0)d_0=d$. If $n:N\hookrightarrow C_0$ is another subobject through which $d_0$ factors after applying $U$, then $m_{0}n$ is among the original supported subobjects. The intersection property gives $r:C_0\to N$ with $m_{0}nr=m_0$. Since $m_0$ is monic, $nr=1_{C_0}$; since $n$ is monic, also $rn=1_N$. Thus every supported subobject of $C_0$ is invertible.

For each member $Q_i$ of the small cogenerating family, considerThis map is injective. If $U(q)d_0=U(q^{\prime })d_0$, preservation of the equalizer of $q,q^{\prime }$ makes $d_0$ factor through the image of that equalizer. Minimality makes its inclusion an isomorphism, so $q=q^{\prime }$.

Write $R_i=\mathcal{D}(D,U{Q}_i)$; it is a set by local smallness of $\mathcal{D}$. Let $S_i\subseteq R_i$ be the image of ${\theta }_i$. For each $s\in S_i$, there is exactly one corresponding $q_{i,s}:C_0\to Q_i$. These maps define the evaluation embedding into cogenerator productsIt is a monomorphism: if $eg=eh$ and $g\ne h$, cogeneration supplies some $q:C_0\to Q_i$ distinguishing $g,h$; that $q$ is one of the projections of $e$, a contradiction. The product is small. It is important to use the subfamilies $S_i$, since some missing coordinate in $R_i$ need not correspond to a morphism out of $C_0$.

There are only a set of possible families $(S_i\subseteq R_i{)}_{i\in I}$. For each such family form $P_S$, choose a set of representatives $N\hookrightarrow P_S$ of its subobjects, and take all pairs $(N,d_N:D\to UN)$. Local smallness of $\mathcal{D}$ and well-poweredness make their union a set. In the case just constructed, $e$ identifies $C_0$ with one chosen representative $N$. Transporting $d_0$ to $UN$ and composing the inverse identification with $m_0$ gives a factorization of the original $d$ through that pair. Therefore these pairs form a weakly initial set in $(D\downarrow U)$.

The general adjoint functor theorem now supplies a left adjoint to $U$. Conversely, a right adjoint preserves small categorical limits, either by the adjunction hom-set bijections and the hom-set detection of categorical limits, or directly from their universal properties. Hence the stated special theorem is proved in both directions.