TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Chosen limits and colimits of a fixed small shape assemble into limit and colimit functors

Statement

Let $J$ be small. If a particular limiting cone is chosen for every $D : J \to C$ , these choices define a functor

$\lim_{J} : [J, C] \to C .$

Chosen colimiting cocones similarly define ${colim}_{J} : [J, C] \to C$ .

Facts & Assumptions

Given: A small $J$ and, for every $D$ , a chosen limit $(L_{D}, λ^{D})$ .

[F1]

Objects and arrows of a functor category are functors and natural transformations (Functor category $[C, D]$ , Natural transformation and its components).

[F2]

A chosen limit supplies a unique factor from every cone (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[F3]

Smallness is cardinality of the indexing category (Assuming Choice, cardinality of a small category and κ-small diagrams).

[L1]

Limit legs are jointly monic (The legs of a limiting cone are jointly monic, and the legs of a colimiting cocone are jointly epic).

[L2]

The colimit construction is the formal dual (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).

Proof

technique · universal property

1.1

For $α : D \Rightarrow E$ , the family $α_{j} λ_{j}^{D} : L_{D} \to E (j)$ is a cone: for $u : j \to k$ , naturality of $α$ and the cone equation give $E (u) α_{j} λ_{j}^{D} = α_{k} D (u) λ_{j}^{D} = α_{k} λ_{k}^{D}$ .

F1F2

2.1

By [F2], there is a unique morphism $\lim α : L_{D} \to L_{E}$ satisfying $λ_{j}^{E} \lim α = α_{j} λ_{j}^{D}$ for every $j$ .

F2step 1.1

3.1

For $1_{D}$ , both $\lim (1_{D})$ and $1_{L_{D}}$ have the same composites with all $λ_{j}^{D}$ ; [L1] gives $\lim (1_{D}) = 1_{L_{D}}$ .

L1step 2.1

3.2

For $D \overset{α}{\Rightarrow} E \overset{β}{\Rightarrow} H$ , the maps $\lim (β α)$ and $(\lim β) (\lim α)$ have composite $β_{j} α_{j} λ_{j}^{D}$ with every $λ_{j}^{H}$ . By [L1] they are equal. Thus the assignments satisfy the functor laws.

L1step 2.1

4.1

The smallness in [F3] ensures that the functor category is used under the library's set-based indexing convention. Reversing every arrow in steps 1.1, 2.1, 3.1, and 3.2 by [L2] gives the chosen-colimit functor.

F3L2step 1.1step 2.1step 3.1step 3.2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 41 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

E. Riehl, Category Theory in Context, Lemma 3.4.6 (standard reference, not scraped)