TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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.

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Small categories, functors, and natural transformations form the strict 2-category $\mathbf{Cat}$

Statement

Small categories, functors, and natural transformations form a strict 2-category $\mathbf{Cat}$ .

Facts & Assumptions

Given: Small categories $\mathcal A,\mathcal B,\mathcal C$ .

[L1]

A strict 2-category has hom-categories, strictly associative and unital horizontal composition, and interchange (Strict 2-category).

[L2]

Functors and natural transformations form functor categories (Functor category $[\mathcal C,\mathcal D]$ ), these hom-categories are small for small endpoints (If $\mathcal C$ is small and $\mathcal D$ is locally small then $[\mathcal C,\mathcal D]$ is locally small; if both are small it is small), and interchange holds (Horizontal and vertical composition of natural transformations satisfy the interchange law).

Proof

technique · direct

1.1

Take small categories as objects and $[\mathcal A,\mathcal B]$ as each hom-category; its objects are functors and its morphisms are natural transformations.

givenL1L2

2.1

Functor composition gives horizontal composition, whiskering gives its action on natural transformations, and ordinary composition of functions makes associativity and units literal equalities.

step 1.1L1L2

3.1

The interchange theorem makes horizontal composition functorial with respect to vertical composition; the class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{CAT}$ is not formed treats the object collection schematically and never forms $\mathbf{CAT}$ , so all strict 2-category axioms hold for $\mathbf{Cat}$ .

step 2.1L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)

Small categories, functors, and natural transformations form the strict 2-category Cat\mathbf{Cat}Cat