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

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The internal hom is unique up to a unique adjunction-compatible natural isomorphism

Statement

Fix an object $X$ in a monoidal category. Any two chosen right adjoints to $- \otimes X$ are related by a unique natural isomorphism compatible with their adjunction units and counits, and the analogous assertion holds for two chosen right adjoints to $X \otimes -$ . Hence each internal-hom construction, together with its adjunction data, is unique up to a unique adjunction-compatible natural isomorphism.

Facts & Assumptions

Given: An object $X$ in a monoidal category, together with two chosen right adjoints to $- \otimes X$ or two chosen right adjoints to $X \otimes -$ .

[L1]

Right-closed and left-closed mean exactly that the corresponding tensor functor has a right adjoint (Left-closed, right-closed, and biclosed monoidal categories).

[L2]

Two right adjoints to the same functor are uniquely naturally isomorphic in a way compatible with the adjunction data (Adjoints are unique up to a unique natural isomorphism compatible with the adjunction data).

Proof

technique · direct

1.1givenL2

If $R, R^{'} : C \to C$ are two chosen right adjoints to $- \otimes X$ , then they are two right adjoints to the same functor. By [L2] there is a unique natural isomorphism $R \Rightarrow R^{'}$ compatible with the adjunction units and counits.

2.1step 1.1L1

By [L1], a right internal hom in the variable $X$ is precisely such a right adjoint $R = [X, -]$ . Therefore any two choices of $[X, -]$ , equipped with their adjunction data, are related by the unique natural isomorphism compatible with those units and counits.

2.2step 1.1L1L2

The same argument with the functor $X \otimes -$ proves the corresponding adjunction-compatible uniqueness for the left internal hom $⟨ X, - ⟩$ .

3.1step 2.1step 2.2∎

So both internal-hom constructions, with their adjunction data fixed, are unique up to a unique compatible natural isomorphism.

Depends on

Used by

Dependency tree · two levels

5 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., Proposition 4.3.1 and Definition 4.4.7 (standard reference, not scraped)
G. M. Kelly, Basic Concepts of Enriched Category Theory, Section 1.5 (standard reference, not scraped)