TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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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.
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A supplied symmetry identifies the left and right internal homs

Statement

Let $C$ be a monoidal category, fix an object $X$ , and suppose there is a natural isomorphism

$σ_{A} : A \otimes X \overset{≅}{⟶} X \otimes A$

natural in $A$ (Natural isomorphism). If $- \otimes X$ has a right adjoint $[X, -]$ , then $X \otimes -$ has a right adjoint uniquely naturally isomorphic to $[X, -]$ . In particular, in a symmetric monoidal category the left and right internal homs of $X$ agree up to unique natural isomorphism.

Facts & Assumptions

Given: A monoidal category, an object $X$ , a natural isomorphism $σ_{A} : A \otimes X ≅ X \otimes A$ , and a chosen right adjoint $[X, -]$ to $- \otimes X$ .

[L1]

The evaluation-transpose bijection for $[X, -]$ is $C (A \otimes X, Y) ≅ C (A, [X, Y])$ (The internal hom and its evaluation morphism).

[L2]

Right adjoints to a fixed functor are unique up to unique natural isomorphism (The internal hom is unique up to a unique adjunction-compatible natural isomorphism).

[L3]

A natural isomorphism is, in particular, a natural family of isomorphisms (Natural isomorphism).

Proof

technique · direct

1.1

For each $A, Y$ , compose the bijection of [L1] with precomposition by $σ_{A}^{- 1} : X \otimes A \to A \otimes X$ . This gives a natural bijection $C (X \otimes A, Y) ≅ C (A, [X, Y])$ .

givenL1L3algebra

2.1

The bijection in step 1.1 says exactly that $[X, -]$ is also a right adjoint to the functor $X \otimes -$ . Hence a left internal hom for $X$ exists and may be chosen to be $[X, -]$ .

step 1.1given

3.1

Any other chosen right adjoint to $X \otimes -$ is uniquely naturally isomorphic to $[X, -]$ by [L2]. Therefore the supplied symmetry identifies the left and right internal homs, and a genuine symmetric monoidal structure gives this conclusion for every $X$ .

step 2.1L2∎

Depends on

Used by

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Sources

G. M. Kelly, Basic Concepts of Enriched Category Theory, Section 1.5 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, 2nd ed., VII.7 (standard reference, not scraped)