= Solution
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>, \b[$U$ 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 $b_i:B\to UA_i$ 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> $W=\prod_iW_i$. 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 bijection
$$
\mathcal A(LB,A)\cong\mathcal B(B,UA).
$$
For a <morphism> $v:B\to B'$, define $Lv$ by $U(Lv)\eta_B=\eta_{B'}v$. Uniqueness proves functoriality and naturality, giving a <left adjoint> $L\dashv U$. 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 gives
$$
|K_f|\leq 2^{2^{|X|}}.
$$
For $X=\varnothing$, 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 \b[the inclusion of compact Hausdorff spaces has a left adjoint], the <compact Hausdorff reflection>:
$$
\boxed{\mathbf{KHaus}(RX,K)\cong\mathbf{Top}(X,JK).}
$$
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 $s=t$. Therefore
$$
\mathcal A(M,Q_i)\hookrightarrow\mathcal B(B,UQ_i),\qquad s\longmapsto U(s)b_M
$$
is an <injective function>. The <evaluation embedding into cogenerator products> embeds $M$ in $\prod_iQ_i^{S_i}$, where $S_i$ is the realized subset of the fixed <set> $\mathcal B(B,UQ_i)$. 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.
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