CorollaryStatement: 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.

Hom(X,−) is continuous, while Hom(−,X) sends every existing small colimit to a limit of sets

Statement

For every object $X$ of a locally small category $C$ , $C (X, -) : C \to S e t$ preserves all small limits that exist. Moreover, for every small diagram $D$ with a colimit, there is a natural bijection

$C (colim D, X) ≅ \lim_{j \in J^{o p}} C (D (j), X) .$

Facts & Assumptions

Given: An object $X$ and the indicated existing limits or colimits.

[L1]

Every covariantly representable functor preserves small limits (Every covariantly representable functor to Set preserves all existing small limits).

[F1]

$C (X, -)$ and $C (-, X)$ are the covariant and contravariant hom-functors (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

[L2]

A colimit in $C$ is a limit in $C^{o p}$ (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).

[F2]

Continuous means preserving all small limits (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

Proof

technique · direct corollary

1.1

The functor $C (X, -)$ is represented by $X$ , so [L1] says that it preserves every small existing limit. By [F2], it is continuous whenever the term is applied to the available small limits of its domain.

L1F1F2

1.2

Regard a colimit cocone $D (j) \to Q$ as a limiting cone in $C^{o p}$ by [L2]. Applying [L1] there to the representable $C^{o p} (X, -) = C (-, X)$ gives the displayed limit of hom-sets.

L1F1L2

2.1

Explicitly, the bijection sends $f : Q \to X$ to the compatible family of composites $D (j) \to Q \to X$ ; the colimit existence and uniqueness clauses give its inverse and prove uniqueness.

step 1.2∎

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: 31 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

E. Riehl, Category Theory in Context, Theorems 3.5.5 and 3.5.6 (standard reference, not scraped)