PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.

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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

1.1

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

1.2

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.

1.3

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.

2.1

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

step 1.2step 1.3L1∎

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: 7 results over 4 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

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)