TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16‡ 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 are adjoint to the diagonal functor

Statement

Let $J$ be small and let $Δ : C \to [J, C]$ be the diagonal functor.

If a limiting cone is supplied for every $D : J \to C$ , the resulting limit functor satisfies $Δ ⊣ \lim_{J}$ .
If a colimiting cocone is supplied for every such $D$ , the resulting colimit functor satisfies ${colim}_{J} ⊣ Δ$ .

The choices are part of the hypotheses.

Facts & Assumptions

Given: The small category $J$ and the supplied choices in the Statement.

[F1]

Chosen limiting or colimiting cones for every diagram assemble into limit or colimit functors (Chosen limits and colimits of a fixed small shape assemble into limit and colimit functors).

[F2]

A limit is a terminal cone and a colimit is an initial cocone (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[F3]

Morphisms in a functor category are natural transformations (Functor category $[C, D]$ ).

Proof

technique · direct

1.1

By [F1], the supplied limiting cones define a functor $\lim_{J} : [J, C] \to C$ .

1.2

A morphism $c \to \lim D$ is, by [F2], uniquely equivalent to a cone from $c$ to $D$ ; by [F3], such a cone is exactly a natural transformation $Δ c \Rightarrow D$ .

F2F3

1.3

Dually, supplied colimits assemble by [F1], and [F2] with [F3] make a morphism $colim D \to c$ uniquely equivalent to a cocone $D \Rightarrow Δ c$ .

F1F2F3

2.1

The correspondence in step 1.2 is natural in $c$ and $D$ because postcomposition of a mediating map and transport of a cone along a natural transformation preserve the defining cone equations. Hence $Δ ⊣ \lim_{J}$ .

step 1.1step 1.2

3.1

Reading step 2.1 in the opposite categories gives the corresponding check for step 1.3: the cocone correspondence is natural in $c$ and $D$ because precomposition of a mediating map and transport of a cocone along a natural transformation preserve the defining cocone equations. Hence ${colim}_{J} ⊣ Δ$ .

step 1.3step 2.1F2F3

4.1

Both constructions begin with supplied objectwise choices; no selection is inferred from bare existence.

step 2.1step 3.1∎

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 11 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, 2nd ed., Proposition 4.6.1 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Section 5.1 (standard reference, not scraped)