TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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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

1.1

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 $G (F (a)) η_{c} = η_{c^{'}} a$ .

F1construct

2.1

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.

step 1.1F1

3.1

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].

step 1.1step 2.1L1

4.1

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.

step 1.1F1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 11 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

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)