TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Under local smallness, representable functors give a second proof that right adjoints preserve small limits

Statement

Let $F ⊣ G : D \to C$ be an adjunction between locally small categories. For every small diagram $D : J \to D$ with a limit, the representable-functor calculation identifies $G (\lim D)$ as a limit of $G D$ . Equivalently, the canonical comparison

$G (\lim D) ⟶ \lim (G D)$

is an isomorphism whenever the displayed chosen limits are supplied.

Facts & Assumptions

Given: The locally small adjunction and small diagram in the Statement.

[F1]

A covariantly representable Set-valued functor on a locally small category preserves every small limit that exists (Every covariantly representable functor to Set preserves all existing small limits).

[F2]

A functor preserves a chosen limit exactly when its canonical comparison to the chosen limit of the image diagram is an isomorphism (A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits).

[L1]

The adjunction gives natural bijections $C (c, G d) ≅ D (F c, d)$ (Under local smallness, transposition gives the natural hom-set bijection, and conversely).

Proof

technique · direct

1.1L1

For every $c \in C$ , [L1] gives $C (c, G \lim D) ≅ D (F c, \lim D)$ .

2.1step 1.1F1

By [F1], the right side is naturally isomorphic to $\lim_{j} D (F c, D_{j})$ , since $D (F c, -)$ is representable and $J$ is small.

3.1step 2.1L1

Applying [L1] componentwise identifies this limit with $\lim_{j} C (c, G D_{j})$ . The composite is natural in $c$ .

4.1step 3.1F2∎

Hence the cone $G (\lim D) \to G D$ represents the cone functor and is limiting. If a chosen limit of $G D$ is also supplied, uniqueness of limits makes the canonical comparison an isomorphism, exactly as stated in [F2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Section 4.6 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Section 6.3 (standard reference, not scraped)