Small-limit preservation is insufficient for the large elements-colimit criterion

Let $\mathcal{C}$ be the ordered category of all ordinals with an added greatest object $\infty$. The categorical presheaf with $X(\alpha )=\{\ast \}$ at ordinals and $X(\infty )=\varnothing$ preserves every small categorical limit: a small supremum is an ordinal unless its diagram contains $\infty$. Its category of elements consists of all ordinals, and their projection has the large colimit $\infty$. Nevertheless $X$ is not representable: an ordinal representer has bounded support, while the representer $\infty$ has a nonempty value at $\infty$. Thus preservation must include the possibly large opposite elements diagram. The dual functor $X^{\mathrm{op}}:\mathcal{C}\to {\mathbf{Set}}^{\mathrm{op}}$ also disproves the comma-projection criterion for a right adjoint under small-colimit preservation alone.