CorollaryStatement: AI-generatedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

A second proof that adjoints are unique

Statement

If an endofunctor $F$ has two left adjoints $L_{1}$ and $L_{2}$ , then there is a unique natural isomorphism $L_{1} ≅ L_{2}$ compatible with the two adjunction structures. The corresponding statement for right adjoints is also true.

Facts & Assumptions

Given: Endofunctors $L_{1}, L_{2}, F : C \to C$ and adjunctions $L_{1} ⊣ F$ and $L_{2} ⊣ F$ .

[L1]

An adjunction to $F$ is the same thing as a dual object of $F$ in the composition monoidal category (A dual object in the endofunctor category is an adjoint functor).

[L2]

Duals of a fixed object are unique up to a unique compatible isomorphism (Duals are unique up to a unique compatible isomorphism).

Proof

technique · direct

1.1

By [L1], the two adjunctions $L_{1} ⊣ F$ and $L_{2} ⊣ F$ make $L_{1}$ and $L_{2}$ into two left duals of the same object $F$ in the endofunctor composition monoidal category.

givenL1

2.1

Applying [L2] to those two duals yields a unique compatible isomorphism $L_{1} ≅ L_{2}$ . Compatibility with the duality data is exactly compatibility with the units and counits of the two adjunctions by [L1].

step 1.1L1L2

3.1

Hence left adjoints of a fixed functor are unique up to unique compatible natural isomorphism. The right-adjoint statement is the same argument with right duals.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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, Proposition 4.2.4 (standard reference, not scraped)
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Proposition 2.10.5 (standard reference, not scraped)