CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.

A right adjoint is left exact and a left adjoint is right exact

Statement

Every right adjoint is left exact, and every left adjoint is right exact. The assertion concerns finite limits or colimits that exist in the source category; it does not assert that the source has them.

Facts & Assumptions

Given: An adjunction $F ⊣ G$ .

[L1]

Right adjoints preserve every limit that exists (Right adjoints preserve every limit that exists).

[L2]

Left adjoints preserve every colimit that exists (Left adjoints preserve every colimit that exists).

[L3]

Left exact means preserving existing finite limits, and right exact means preserving existing finite colimits (Left exact and right exact functors).

Proof

technique · direct

1.1

Applying [L1] to finite indexing categories shows that $G$ preserves every existing finite limit.

1.2

Applying [L2] to finite indexing categories shows that $F$ preserves every existing finite colimit.

2.1

Unfolding [L3], step 1.1 says that $G$ is left exact and step 1.2 says that $F$ is right exact, without adding an existence hypothesis.

step 1.1step 1.2L3∎

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: 18 results over 7 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., Definition 4.6.7 and Theorem 4.6.2 (standard reference, not scraped)