= Limit form of the special adjoint functor theorem
A <functor> $G:\mathcal C\to\mathcal D$ from a <locally small category> that is complete and <well-powered>, with a <small cogenerating family>, to a <locally small category> has a <left adjoint> exactly when it preserves small <categorical limits>. For sufficiency, the <cogenerator bound for comma-category solution sets> produces a <weakly initial set> in each complete <comma category> $(B\downarrow G)$. The <initial-object lemma for complete categories with a weakly initial set> supplies its <initial object>, a <universal arrow from an object to a functor>. Necessity is limit preservation by a <right adjoint>.
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