DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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 internal hom and its evaluation morphism

Definition

Assume $C$ is right closed. For objects $X, Y$ , the right internal hom $[X, Y]$ is the value at $Y$ of the chosen right adjoint $[X, -]$ to $- \otimes X$ (Left-closed, right-closed, and biclosed monoidal categories).

The evaluation morphism

${ev}_{X, Y} : [X, Y] \otimes X \to Y$

is the counit component at $Y$ of the adjunction $- \otimes X ⊣ [X, -]$ . For every object $A$ , the adjunction yields the transposition bijection

$C (A \otimes X, Y) ≅ C (A, [X, Y]),$

which sends a morphism $u : A \otimes X \to Y$ to its adjunct $u^{♭} : A \to [X, Y]$ and sends $v : A \to [X, Y]$ to its inverse transpose $v^{♯} : A \otimes X \to Y$ in the sense of Adjuncts and transposition under an adjunction.

If $C$ is left closed, the left internal hom $⟨ X, Y ⟩$ and its evaluation map

${\overset{}{ev}}_{X, Y} : X \otimes ⟨ X, Y ⟩ \to Y$

are defined dually from the adjunction $X \otimes - ⊣ ⟨ X, - ⟩$ .

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

G. M. Kelly, Basic Concepts of Enriched Category Theory, equations (1.23) and (1.24) (standard reference, not scraped)
Emily Riehl, Category Theory in Context, 2nd ed., Section 4.4 (standard reference, not scraped)