TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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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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Lan is left adjoint to restriction, and restriction is left adjoint to Ran

Statement

Let $K : C \to D$ be a functor with $C$ and $D$ small and $E$ locally small, so that the restriction functor

$K^{*} : [D, E] ⟶ [C, E]$

is defined (Global Kan extensions as adjoints to restriction).

Assume that for every functor $F : C \to E$ a local left Kan extension of $F$ along $K$ is supplied, and for every such $F$ a local right Kan extension of $F$ along $K$ is supplied.

Then the object assignments $F \mapsto {Lan}_{K} F$ and $F \mapsto {Ran}_{K} F$ admit unique functor structures for which

${Lan}_{K} ⊣ K^{*} ⊣ {Ran}_{K} .$

No class-indexed choice is made inside the proof: the local Kan extensions are part of the data.

Facts & Assumptions

Given: The functor $K : C \to D$ with $C, D$ small and $E$ locally small; the restriction functor $K^{*}$ ; and supplied local left and right Kan extensions for every $F : C \to E$ .

[F1]

A local left Kan extension of $F$ along $K$ is a pair $(L, η)$ initial among natural transformations $F \Rightarrow M K$ , and a local right Kan extension is a pair $(R, ε)$ terminal among natural transformations $M K \Rightarrow F$ (Left and right Kan extensions).

[L1]

Chosen universal arrows from each object to a functor assemble uniquely into a left adjoint (Chosen objectwise universal arrows assemble uniquely into a left adjoint).

[L2]

A left adjoint to a functor is supplied exactly by chosen initial objects in its comma categories, and dually a right adjoint is supplied exactly by chosen terminal objects (A left adjoint exists exactly when chosen initial objects are supplied in every comma category).

Proof

technique · direct

1.1

By [F1], each supplied local left Kan extension of a functor $F : C \to E$ is exactly a universal arrow from the object $F$ of $[C, E]$ to the restriction functor $K^{*}$ , while each supplied local right Kan extension is exactly a terminal object of the corresponding comma category for $K^{*}$ .

2.1

Applying [L1] to the supplied universal arrows of step 1.1 gives a unique functor structure on $F \mapsto {Lan}_{K} F$ for which the displayed unit transformations are natural, and with that structure ${Lan}_{K} ⊣ K^{*}$ .

L1step 1.1

3.1

Applying the dual clause of [L2] to the supplied terminal objects of step 1.1 gives a unique functor structure on $F \mapsto {Ran}_{K} F$ with $K^{*} ⊣ {Ran}_{K}$ . Hence ${Lan}_{K} ⊣ K^{*} ⊣ {Ran}_{K}$ .

L2step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

E. Riehl, Category Theory in Context, 2nd ed., Proposition 6.1.6 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, §4.1 (standard reference, not scraped)