LemmaStatement: 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.

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

1.1L2algebra

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

1.2given

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

1.3L2algebra

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

1.4given

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

2.1step 1.1step 1.2L1

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

3.1step 1.3step 1.4L1∎

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

Depends on

Used by

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, 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)