= Cogenerator bound for comma-category solution sets
{title2=$\mathcal C(C_0,Q_i)\hookrightarrow\mathcal D(B,GQ_i)$}
Under the hypotheses of the <limit form of the special adjoint functor theorem>, intersect all <subobjects> of $C$ supporting $x:B\to GC$. Limit preservation gives a smallest supporting $(C_0,x_0)$. If $G(u)x_0=G(v)x_0$ for maps $u,v:C_0\to Q_i$, their <equalizer> supports $x_0$, so minimality makes it invertible and $u=v$. Thus the indicated map of <hom-sets> is injective. The <evaluation embedding into cogenerator products> embeds $C_0$ in a product of cogenerators indexed by the realized subsets of the fixed sets $\mathcal D(B,GQ_i)$. There are only a set of such products, a set of their <subobjects>, and a set of maps from $B$ into their images under $G$. These data give a <weakly initial set> in $(B\downarrow G)$. Using realized subsets avoids assuming maps into every cogenerator exist.
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