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Let $\emptyset$ be the empty category. Consider the general adjoint functor theorem: it says that if $\mathcal{D}$ is locally small and complete then $G:\mathcal{D} \to \mathcal{C}$ has a left adjoint $\iff$ $G$ preserves all limits and for each object $A$ of $\mathcal{C}$, the category $(A \downarrow G)$ has a weakly initial set.

However the empty functor $G: \emptyset \to \mathcal{C}$ clearly has no left adjoint (for $\mathcal{C}$ some non-empty category). Which hypothesis of the General Adjoint Functor Theorem do we contradict?

Is it not true that $\emptyset$ is locally small, and complete, and $(A \downarrow G)$ has a weakly initial set (i.e. the empty set) and $G$ preserves limits?

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The empty category is not complete: the empty diagram does not have a limit (that is, the category has no terminal object). – Arturo Magidin May 28 '12 at 22:07
Of course. Thanks! – Paul Slevin May 28 '12 at 22:08
up vote 6 down vote accepted

Just to make sure the question does not go without an answer:

The empty category is not complete, because the empty diagram does not have a limit. Such a limit would be a terminal object of the category, which of course the empty category does not have.

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