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.
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Unit components are initial in comma categories, and counit components are terminal

Statement

Let $F ⊣ G$ have unit $η$ and counit $ε$ .

For each $c \in C$ , $(F c, η_{c} : c \to G F c)$ is a universal arrow from $c$ to $G$ , hence an initial object of $(c ↓ G)$ .
For each $d \in D$ , $(G d, ε_{d} : F G d \to d)$ is a universal arrow from $F$ to $d$ , hence a terminal object of $(F ↓ d)$ .

No local-smallness hypothesis is needed.

Facts & Assumptions

Given: An adjunction $F ⊣ G$ with unit $η$ and counit $ε$ .

[F1]

A universal arrow from $c$ to $G$ is a pair $(R, ρ : c \to G R)$ such that every $f : c \to G d$ factors uniquely as $G (h) \circ ρ$ ; dually, a universal arrow from $F$ to $d$ has the corresponding unique factorisation property (Universal arrows from an object to a functor and from a functor to an object).

[F2]

A universal arrow from an object to a functor is initial in the associated comma category, and a universal arrow from a functor to an object is terminal in the dual comma category (Universal arrows to a functor are initial in comma categories, and universal arrows from a functor are terminal).

[L1]

The formulas $h \mapsto G (h) η_{c}$ and $f \mapsto ε_{d} F (f)$ are mutually inverse for an adjunction (The unit and counit transpose formulas are mutually inverse, Adjuncts and transposition under an adjunction).

Proof

technique · direct

1.1

Fix $c$ and a morphism $f : c \to G d$ . Its inverse transpose $f^{♯} : F c \to d$ satisfies $G (f^{♯}) \circ η_{c} = f$ , and it is the unique morphism with this property because transposition is injective.

1.2

Dually, for $u : F c \to d$ , its transpose $u^{♭} : c \to G d$ is the unique morphism satisfying $ε_{d} \circ F (u^{♭}) = u$ .

2.1

Thus $(F c, η_{c})$ has the universal property in [F1], and [F2] makes it initial in $(c ↓ G)$ .

step 1.1F1F2

3.1

Hence $(G d, ε_{d})$ is universal from $F$ to $d$ and terminal in $(F ↓ d)$ . The argument used individual morphisms and uniqueness only, so it imposed no set-size condition on any hom-class.

step 1.2F1F2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 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

Emily Riehl, Category Theory in Context, 2nd ed., Theorem 4.2.7 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Theorem 2.3.6 (standard reference, not scraped)