TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Right enriched adjoints preserve weighted limits

Statement

If $F : A \to B$ is left adjoint to $G : B \to A$ in the enriched sense, then $G$ preserves every enriched weighted limit that exists in $B$ .

Facts & Assumptions

Given: An enriched adjunction $F ⊣_{V} G$ and a weighted limit ${W, T}$ in $B$ .

[L1]

An enriched adjunction is a natural isomorphism $B (F A, B) ≅ A (A, G B)$ (Enriched adjunction).

[L2]

A weighted limit represents the enriched natural-transformation object against the hom-functor (Enriched weighted limit).

Proof

technique · direct

1.1

For each $A \in A$ , apply [L1] with $B = {W, T}$ to identify $A (A, G {W, T})$ with $B (F A, {W, T})$ .

L1given

2.1

Because ${W, T}$ is a weighted limit, [L2] identifies $B (F A, {W, T})$ with the enriched transformation object $[J, V] (W, B (F A, T -))$ . Using [L1] again pointwise in the diagram variable replaces $B (F A, T -)$ by $A (A, G T -)$ .

L1L2step 1.1

3.1

Step 2.1 is exactly the representing property for $G {W, T}$ as the weighted limit of $G T$ . Therefore $G$ preserves the weighted limit.

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

G. M. Kelly, Basic Concepts of Enriched Category Theory, Section 3.2 (standard reference, not scraped)