LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

The unit and counit transpose formulas are mutually inverse

Statement

Let $F ⊣ G$ have unit $η$ and counit $ε$ . For every $u : F c \to d$ and $v : c \to G d$ ,

$(u^{♭})^{♯} = u, (v^{♯})^{♭} = v,$

where transposition is defined in Adjuncts and transposition under an adjunction.

Facts & Assumptions

Given: An adjunction $F ⊣ G$ with unit $η$ and counit $ε$ , objects $c, d$ , and morphisms $u : F c \to d$ and $v : c \to G d$ .

[L1]

The triangle identities are $ε_{F c} \circ F (η_{c}) = 1_{F c}$ and $G (ε_{d}) \circ η_{G d} = 1_{G d}$ (Adjunction by unit, counit, and the triangle identities).

[L2]

The transpose formulas are $u^{♭} = G (u) \circ η_{c}$ and $v^{♯} = ε_{d} \circ F (v)$ (Adjuncts and transposition under an adjunction).

Proof

technique · direct

1.1

Expanding the first composite gives $(u^{♭})^{♯} = ε_{d} \circ F G (u) \circ F (η_{c})$ .

L2algebra

1.2

Naturality of $ε$ at $u$ gives $ε_{d} \circ F G (u) = u \circ ε_{F c}$ .

given

1.3

Expanding the second composite gives $(v^{♯})^{♭} = G (ε_{d}) \circ G F (v) \circ η_{c}$ .

L2algebra

1.4

Naturality of $η$ at $v$ gives $G F (v) \circ η_{c} = η_{G d} \circ v$ .

given

2.1

Substituting step 1.2 into step 1.1 and applying the first triangle identity yields $(u^{♭})^{♯} = u \circ ε_{F c} \circ F (η_{c}) = u$ .

step 1.1step 1.2L1

3.1

Substituting step 1.4 into step 1.3 and applying the second triangle identity yields $(v^{♯})^{♭} = G (ε_{d}) \circ η_{G d} \circ v = v$ .

step 1.3step 1.4L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 7 results over 6 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, Lemmas 2.2.2 and 2.2.4 (standard reference, not scraped)
Emily Riehl, Category Theory in Context, 2nd ed., Theorem 4.2.7 (standard reference, not scraped)