DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Enriched natural transformation

Definition

Let $T, S : A \to B$ be $V$ -functors (Enriched functor).

A $V$ -natural transformation $α : T \Rightarrow S$ is a family of morphisms in $V$

$α_{A} : 1 ⟶ B (T A, S A)$

indexed by the objects $A$ of $A$ , such that for every pair $A, B$ the following two composites from $A (A, B) ≅ 1 \otimes A (A, B)$ to $B (T A, S B)$ agree:

$M_{T A, S A, S B}^{B} \circ (B (S A, S B) \otimes α_{A}) \circ (S_{A, B} \otimes 1),$

$M_{T A, T B, S B}^{B} \circ (α_{B} \otimes B (T A, T B)) \circ (1 \otimes T_{A, B}),$

with the unitors inserted in the evident way to identify $A (A, B)$ with $1 \otimes A (A, B)$ and with $A (A, B) \otimes 1$ .

Equivalently, once the convention of The order of the tensor factors in enriched composition is fixed on this page is fixed, the same naturality law may be rewritten in the compact square form proved later on this page.

Depends on

Enriched functor

Used by

Dependency tree · two levels

2 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, equation (1.7) (standard reference, not scraped)
Geoffrey Cruttwell, Normed Spaces and the Change of Base for Enriched Categories, Section 2.2.2 (standard reference, not scraped)