= Solution
The <right-adjoint criterion using comma-category colimits> is as follows. Under the given colimit-preservation hypothesis, with preservation including the possibly large colimits below, $F:\mathcal X\to\mathcal A$ has a <right adjoint> exactly when, for every $A$, the projection
$$
U_A:(F\downarrow A)\to\mathcal X,\qquad (X,f:FX\to A)\longmapsto X
$$
has a <colimit> in $\mathcal X$. \b[The right adjoint is obtained from these comma-category colimits.] Thus the objects to construct are
$$
\boxed{GA=\operatorname{colim}_{(X,f)\in(F\downarrow A)}X.}
$$
For necessity, if $F\dashv G$ with <adjunction counit> $\varepsilon$, $(GA,\varepsilon_A)$ is a <terminal object> of the <comma category> $(F\downarrow A)$. Its unique incoming <morphisms> give a <colimit> cocone for $U_A$, exactly as for the elements projection in the preceding part.
For sufficiency, choose such a <colimit> $L$ with legs $u_{X,f}:X\to L$. The <morphisms> $f:FX\to A$ are a <cocone> on $FU_A$. Since $F$ preserves this <colimit>, there is a unique $\varepsilon_A:FL\to A$ satisfying
$$
\varepsilon_A F(u_{X,f})=f.
$$
Each $u_{X,f}$ is thus a <morphism> $(X,f)\to(L,\varepsilon_A)$ in the <comma category>. The original <colimit> cocone gives $u_{L,\varepsilon_A}u_{X,f}=u_{X,f}$ for every object, hence $u_{L,\varepsilon_A}=1_L$ by the <universal property>. For any other <morphism> $h:(X,f)\to(L,\varepsilon_A)$, cocone compatibility gives
$$
u_{X,f}=u_{L,\varepsilon_A}h=h.
$$
So $(L,\varepsilon_A)$ is a <terminal object>. Equivalently, $L$ represents the <categorical presheaf> $\mathcal A(F-,A)$, via $h\mapsto\varepsilon_A Fh$. The <functoriality of chosen representations> makes these $L$ into $G:\mathcal A\to\mathcal X$, yielding a <natural bijection>
$$
\boxed{\mathcal X(X,GA)\cong\mathcal A(FX,A),\qquad F\dashv G.}
$$
The preservation of these possibly large <colimits> is essential to the construction of $\varepsilon_A$. Under a small-only interpretation, the preceding <ordinal> counterexample also disproves the unqualified converse here. Take $F=X^{\mathrm{op}}:\mathcal C\to\mathbf{Set}^{\mathrm{op}}$. It preserves all small <colimits>. For a nonempty <set> $A$, the <comma category> $(F\downarrow A)$ has one object over each <ordinal> and none over $\infty$, so its projection has <colimit> $\infty$. For $A=\varnothing$, the projection is the identity of $\mathcal C$, again with <colimit> $\infty$. Thus all these projection colimits exist, but $F$ has no <right adjoint>: the <categorical presheaf> $\mathbf{Set}^{\mathrm{op}}(F-,\{*\})=X$ is not <representable>. This makes the large-preservation qualification substantive.
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