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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The underlying-category construction is a 2-functor

Statement

Let $V$ be locally small. Sending a $V$ -category $A$ to its underlying ordinary category $A_{0}$ extends to enriched functors and enriched natural transformations and defines a strict 2-functor $(-)_{0} : V - C a t \to C a t$ on the set-object enriched categories of this page.

Facts & Assumptions

Given: A locally small monoidal category $V$ , $V$ -categories $A, B$ , and $V$ -functors $T, S : A \to B$ .

[L1]

The underlying category $A_{0}$ has the same objects as $A$ , hom-sets $V (1, A (A, B))$ , and composition induced from enriched composition (The underlying ordinary category of an enriched category).

[L2]

A $V$ -functor gives hom-object maps $T_{A, B} : A (A, B) \to B (T A, T B)$ compatible with composition and units (Enriched functor).

[L3]

A $V$ -natural transformation has components $α_{A} : 1 \to B (T A, S A)$ (Enriched natural transformation).

Proof

technique · direct

1.1

On objects, define $T_{0} : A_{0} \to B_{0}$ by the same object map as $T$ . On a morphism $f : 1 \to A (A, B)$ , define $T_{0} (f) : = T_{A, B} \circ f : 1 \to B (T A, T B)$ . Because [L2] preserves enriched composition and identities, [L1] shows that $T_{0}$ preserves ordinary composition and identity morphisms.

L1L2given

1.2

On 2-cells, send $α : T \Rightarrow S$ to the ordinary natural transformation $α_{0} : T_{0} \Rightarrow S_{0}$ whose component at $A$ is exactly the same morphism $α_{A} : 1 \to B (T A, S A)$ from [L3]. The enriched naturality equation of [L3] implies ordinary naturality after reading the hom-objects through [L1].

L1L3algebra

2.1

Identity functors, composite functors, identity 2-cells, and vertical and horizontal composites are preserved strictly because the definitions in steps 1.1 and 1.2 forget no object-level data and simply reuse the same component maps. Therefore the assignment is a strict 2-functor.

step 1.1step 1.2∎

Depends on

Used by

The underlying ordinary category is change of base along the underlying-hom functor Corollary

Dependency tree · two levels

4 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, equations (1.12) and (1.13) (standard reference, not scraped)
Emily Riehl, Categorical Homotopy Theory, Remark 3.5.11 (standard reference, not scraped)