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?