= Solution
Take the <category of ordinals in reverse order> $\mathbf{Ord}^{\mathrm{op}}$, with a unique arrow $\alpha\to\beta$ exactly when $\alpha\geq\beta$, and the unique <functor>
$$
G:\mathbf{Ord}^{\mathrm{op}}\longrightarrow\mathbf1.
$$
Both categories are <complete categories>. A small limit in the ordinal category is the supremum of the ordinals occurring in the diagram; the empty supremum is $0$, its <terminal object>. The functor $G$ preserves all these <categorical limits>, and both categories are <locally small categories>.
A <left adjoint> would select an <initial object> of $\mathbf{Ord}^{\mathrm{op}}$, meaning an ordinal at least as large as every ordinal. No such object exists. More specifically, any set of candidate ordinals has an upper bound $\gamma$, and none of its members maps to $\gamma+1$. Thus \b[the solution-set condition fails]. This example isolates the genuine smallness requirement in the <general adjoint functor theorem>.
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