PropositionStatement: AI-adaptedProof: AI-generatedprecheck 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 $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small

Statement

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

Facts & Assumptions

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

[L1]

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

[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

1.1givenL1L2

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

2.1step 1.1L1L2

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

3.1step 2.1L1L2∎

If $D$ 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 $[C, D]$ is small.

Depends on

Dependency tree · 2 levels

Sources

Emily Riehl, Category Theory in Context, Chapter 1

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