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

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A dual object in the endofunctor category is an adjoint functor

Statement

Let $C$ be a category and write tensor product in its endofunctor category as composition. Then a left dual of an endofunctor $F : C \to C$ is exactly a left adjoint of $F$ , and a right dual of $F$ is exactly a right adjoint of $F$ .

Facts & Assumptions

Given: Endofunctors $L, F, R : C \to C$ .

[L1]

A left dual of $F$ consists of natural transformations $ev : L \circ F \Rightarrow 1_{C}$ and $coev : 1_{C} \Rightarrow F \circ L$ satisfying the two zig-zag identities (The zig-zag identities).

[L2]

An adjunction $L ⊣ F$ is a unit $1_{C} \Rightarrow F \circ L$ and counit $L \circ F \Rightarrow 1_{C}$ satisfying the two triangle identities (Adjunction by unit, counit, and the triangle identities).

Proof

technique · direct

1.1

Under the tensor-by-composition convention, the data named in [L1] and [L2] are literally the same pair of natural transformations with the same sources and targets.

givenL1L2

2.1

The two zig-zag identities for a left dual in the composition monoidal structure are exactly the two triangle identities for an adjunction, because both say that the composites $F \overset{coev F}{\Rightarrow} F L F \overset{F ev}{\Rightarrow} F and L \overset{L coev}{\Rightarrow} L F L \overset{ev L}{\Rightarrow} L$ are identities.

step 1.1L1L2

3.1

Therefore $L$ is a left dual of $F$ exactly when $L ⊣ F$ . The same comparison with the mirrored data shows that $R$ is a right dual of $F$ exactly when $F ⊣ R$ .

step 2.1∎

Depends on

Used by

A second proof that adjoints are unique Corollary
An adjunction read as a duality of endofunctors Example

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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Exercise 2.10.4 (standard reference, not scraped)
Emily Riehl, Category Theory in Context, Definition 4.1.1 (standard reference, not scraped)