TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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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.

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Chosen objectwise universal arrows assemble uniquely into a left adjoint

Statement

Let $G : D \to C$ be a functor. Suppose that for every $c \in C$ a universal arrow $(F c, η_{c} : c \to G F c)$ from $c$ to $G$ is supplied. There is a unique functor structure on the object assignment $c \mapsto F c$ for which $η : 1_{C} \Rightarrow G F$ is natural, and with that structure $F ⊣ G$ .

Facts & Assumptions

Given: The functor $G$ and the supplied universal arrows in the Statement.

[F1]

Universality means that every $f : c \to G d$ factors uniquely as $f = G (h) \circ η_{c}$ for a morphism $h : F c \to d$ (Universal arrows from an object to a functor and from a functor to an object).

[L1]

Chosen initial objects $(F c, η_{c})$ in all $(c ↓ G)$ determine a unique left adjoint functor (A left adjoint exists exactly when chosen initial objects are supplied in every comma category).

Proof

technique · direct

1.1F1construct

For $a : c \to c^{'}$ , apply [F1] to $η_{c^{'}} a : c \to G F c^{'}$ and define $F (a)$ as the unique morphism satisfying $G (F (a)) η_{c} = η_{c^{'}} a$ .

2.1step 1.1F1

The uniqueness clause in [F1] gives $F (1_{c}) = 1_{F c}$ and $F (b a) = F (b) F (a)$ , because the proposed right sides satisfy the same defining equations.

3.1step 1.1step 2.1L1

Hence $F$ is a functor and the equations in step 1.1 say exactly that $η$ is natural. The same universal factorisations give the adjunction $F ⊣ G$ by [L1].

4.1step 1.1F1∎

Any other compatible functor structure would have to satisfy the defining equation in step 1.1, so [F1] makes it identical to this one on every morphism.

Depends on

Used by

Dependency tree · two levels

8 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

Tom Leinster, Basic Category Theory, Theorem 2.3.6 (standard reference, not scraped)
Emily Riehl, Category Theory in Context, 2nd ed., Section 4.2 (standard reference, not scraped)