PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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.

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

Statement

If $\mathcal C$ is small and $\mathcal D$ is locally small, then $[\mathcal C,\mathcal D]$ is locally small. If both $\mathcal C$ and $\mathcal D$ are small, then $[\mathcal C,\mathcal D]$ is small.

Facts & Assumptions

Given: Categories $\mathcal C,\mathcal D$ with the stated size hypotheses.

[L1]

The objects and morphisms of $[\mathcal C,\mathcal D]$ are functors and natural transformations (Functor category $[\mathcal C,\mathcal D]$ ).

[L2]

Smallness and local smallness mean set-sized object, morphism, and hom-collections as specified in Small, locally small, and large categories.

Proof

technique · direct

1.1

For fixed functors $F,G$ , a natural transformation is a family in the set-indexed product $\prod_{C\in\operatorname{Ob}\mathcal C}\mathcal D(FC,GC)$ satisfying a set of naturality equations; smallness of $\mathcal C$ and local smallness of $\mathcal D$ make this a set.

givenL1L2

2.1

Hence every hom-collection of $[\mathcal C,\mathcal D]$ is a set, so the functor category is locally small.

step 1.1L1L2

3.1

If $\mathcal D$ is also small, the possible object and morphism functions of a functor lie in set-sized function spaces, and the functor equations define a subset; the union of the set-sized natural-transformation sets is then a set, so $[\mathcal C,\mathcal D]$ is small.

step 2.1L1L2∎

Depends on

Used by

Small categories, functors, and natural transformations form the strict 2-category Cat Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 8 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)

If C\mathcal CC is small and D\mathcal DD is locally small then [C,D][\mathcal C,\mathcal D][C,D] is locally small; if both are small it is small