TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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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The free-monoid functor is left adjoint to the underlying-set functor

Statement

The assignment $X \mapsto X^{*}$ of the finite-word monoid extends to a functor $(-)^{*} : S e t \to M o n$ and is left adjoint to the underlying-set functor $U : M o n \to S e t$ .

Facts & Assumptions

Given: A set $X$ and its finite-word monoid $X^{*}$ .

[L1]

The one-letter map $i_{X} : X \to X^{*}$ is universal: every function $X \to U (M)$ extends uniquely to a monoid homomorphism $X^{*} \to M$ (Finite words satisfy the free-monoid universal property).

[L2]

Chosen objectwise universal arrows assemble uniquely into a left adjoint (Chosen objectwise universal arrows assemble uniquely into a left adjoint).

Proof

technique · direct

1.1

For a function $a : X \to Y$ , let $a^{*} : X^{*} \to Y^{*}$ be the unique monoid homomorphism extending the one-letter function $i_{Y} a$ .

L1construct

1.2

Restriction to letters and the extension in [L1] are inverse, naturally identifying monoid homomorphisms $X^{*} \to M$ with functions $X \to U (M)$ .

2.1

Uniqueness in [L1] gives $(1_{X})^{*} = 1_{X^{*}}$ and $(b a)^{*} = b^{*} a^{*}$ , since each pair agrees on all one-letter words. Thus $(-)^{*}$ is a functor and $i_{X}$ is natural.

step 1.1L1

3.1

Therefore $(X^{*}, i_{X})$ is a universal arrow from $X$ to $U$ , and [L2] gives $(-)^{*} ⊣ U$ , including the empty-set case.

step 1.1step 2.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.10 (standard reference, not scraped)