ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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An adjunction read as a duality of endofunctors

Example

Fix a set $A$ . In the endofunctor category of $S e t$ under composition, the functor $(-) \times A$ has right dual the exponential functor $(-)^{A}$ .

Facts & Assumptions

Given: A set $A$ and the endofunctors $(-) \times A$ and $(-)^{A}$ of $S e t$ .

[L1]

Currying gives an adjunction $(-) \times A ⊣ (-)^{A}$ (Currying gives the adjunction $- \times A ⊣ (-)^{A}$ in $S e t$ ).

[L2]

In the composition monoidal category, right adjoints are exactly right duals (A dual object in the endofunctor category is an adjoint functor).

Verification

technique · direct

1.1

The currying theorem Currying gives the adjunction $- \times A ⊣ (-)^{A}$ in $S e t$ says exactly that $(-) \times A ⊣ (-)^{A}$ .

givenL1

2.1

By A dual object in the endofunctor category is an adjoint functor, a right adjoint of an endofunctor is exactly its right dual in the composition monoidal category. Therefore $(-)^{A}$ is the right dual of $(-) \times A$ .

step 1.1L2

3.1

So this ordinary adjunction is literally an instance of categorical duality in an endofunctor monoidal category.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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, Section 4.1 (standard reference, not scraped)