DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Adjuncts and transposition under an adjunction

Definition

Let $F ⊣ G : D \to C$ be an adjunction in the sense of Adjunction by unit, counit, and the triangle identities, with unit $η$ and counit $ε$ . For objects $c \in C$ and $d \in D$ :

the right adjunct, or transpose, of $u : F c \to d$ is $u^{♭} : = G (u) \circ η_{c} : c \to G d;$
the left adjunct, or inverse transpose, of $v : c \to G d$ is $v^{♯} : = ε_{d} \circ F (v) : F c \to d .$

The symbols $(-)^{♭}$ and $(-)^{♯}$ always refer to the displayed adjunction; when several adjunctions occur, the relevant unit and counit are named explicitly.

Depends on

Adjunction by unit, counit, and the triangle identities

Used by

Dependency tree · two levels

3 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

Emily Riehl, Category Theory in Context, 2nd ed., Section 4.2 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Lemma 2.2.4 (standard reference, not scraped)