Right-adjoint criterion using comma-category colimits
= Right-adjoint criterion using comma-category colimits
{title2=$GA=\operatorname{colim}U_A$}
A <functor> $F:\mathcal X\to\mathcal A$ preserving the possibly large colimits of the projections $U_A:(F\downarrow A)\to\mathcal X$ has a <right adjoint> exactly when all those <colimits> exist. The universal cocone makes its vertex a <terminal object> of $(F\downarrow A)$, hence a representation of $\mathcal A(F-,A)$.