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On a complete category that is locally small, a set-valued functor is representable exactly when it preserves small categorical limits and its category of elements has a weakly initial set. Categorical limit preservation makes that comma category complete. The initial-object lemma for complete categories with a weakly initial set then supplies a universal element, hence a representation of a functor.

Suppose a small category has equalizers. If a categorical presheaf $P$ is a retract $P\stackrel{i}{\to }{H}_B\stackrel{p}{\to }P$, with $pi=1_P$, then $i$ equalizes $ip$ and $1_{H_B}$. The Yoneda lemma writes $ip=H_b$ for an endomorphism $b:B\to B$. An equalizer $A\to B$ of $b$ and $1_B$ gives the same presheaf equalizer, since the Yoneda embedding preserves categorical limits. Uniqueness of equalizers gives $P\cong H_A$.

A functor $F:I\to J$ is initial if restriction preserves categorical limits for diagrams in a category with codomain ${\mathbf{Set}}^{\mathrm{op}}$. To test necessity, view $J(-,j)$ as such a diagram. Its limit is a singleton, since the colimit of the corresponding representable presheaf is the connected-component set of the slice $J\downarrow j$, which has a terminal object. The restricted colimit is the connected-component set of $(F\downarrow j)$. Thus that comma category is nonempty and connected. Sufficiency follows from cone restriction along an initial functor.

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.