PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

In a poset adjunction the triangle identities are automatic

Statement

Let $F : A \to B$ and $G : B \to A$ be monotone maps between posets. If there are natural transformations $η : 1_{A} \Rightarrow G F$ and $ε : F G \Rightarrow 1_{B}$ , equivalently the pointwise inequalities

$a \leq G F (a), F G (b) \leq b,$

then both triangle identities hold automatically. In particular, the unit and counit inequalities of a Galois connection determine an adjunction without a separate triangle calculation.

Facts & Assumptions

Given: The monotone maps and pointwise inequalities in the Statement.

[F1]

A preorder determines a category with at most one morphism between any two objects, and functors between such categories are exactly monotone maps (A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps).

[L1]

A Galois connection supplies the unit inequalities $a \leq G F (a)$ and counit inequalities $F G (b) \leq b$ (Galois connection between preorders).

Proof

technique · direct

1.1F1

By [F1], each pointwise inequality is the unique possible morphism with its source and target, so the supplied families are natural transformations.

1.2F1

At $a \in A$ , the two sides of the first triangle identity are parallel morphisms $F (a) \to F (a)$ in the thin category $B$ , hence they are equal.

1.3F1

At $b \in B$ , the two sides of the second triangle identity are parallel morphisms $G (b) \to G (b)$ in the thin category $A$ , hence they are equal.

2.1step 1.2step 1.3L1∎

Thus both triangle identities hold. Applying this to the inequalities in [L1] proves the final assertion.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tom Leinster, Basic Category Theory, Example 2.2.7 (standard reference, not scraped)
Emily Riehl, Category Theory in Context, 2nd ed., Section 4.2 (standard reference, not scraped)