TheoremStatement: AI-adaptedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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The comparison functor to the Eilenberg–Moore category exists and is unique

Statement

Let $F : C ⇄ D : U$ be an adjunction with counit $ε$ inducing a fixed monad $T$ on $C$ on the nose, and write $F^{T} ⊣ U^{T}$ for the Eilenberg–Moore adjunction with counit $ε^{T}$ . There is exactly one functor $K : D \to C^{T}$ satisfying

$U^{T} K = U, K F = F^{T}, K (ε_{d}) = ε_{K d}^{T} for every object d,$

namely the comparison functor

$K (d) = (U d, U ε_{d}), K (h) = U (h) .$

These three equalities are what it means for $K$ to be a morphism of adjunctions from $F ⊣ U$ to the Eilenberg–Moore adjunction.

Facts & Assumptions

Given: An adjunction $F ⊣ U$ with unit $η$ and counit $ε$ as in Adjunction by unit, counit, and the triangle identities, inducing $T = U F$ and $μ = U ε F$ as in Every adjunction induces a monad on the domain of its left adjoint, together with the Eilenberg–Moore adjunction of The free–forgetful Eilenberg–Moore adjunction induces the given monad.

Proof

technique · direct

1.1

For $d \in D$ set $K (d) = (U d, U ε_{d})$ , and for $h : d \to d^{'}$ set $K (h) = U h$ .

given

2.1

The triangle identity gives $U ε_{d} \circ η_{U d} = 1_{U d}$ . Naturality of $ε$ at $ε_{d}$ gives the algebra associativity equation for $U ε_{d}$ , and naturality at $h$ gives $U h \circ U ε_{d} = U ε_{d^{'}} \circ T (U h)$ ; hence the objects and arrows in step 1.1 are algebras and algebra homomorphisms.

step 1.1given

3.1

Directly $U^{T} K = U$ , and $K F A = (U F A, U ε_{F A}) = (T A, μ_{A}) = F^{T} A$ ; the Eilenberg–Moore counit at $K (d)$ is $U ε_{d}$ , so the counit data also agree. These strict equalities force the underlying arrow action and every structure map, proving uniqueness.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 10 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

E. Riehl, Category Theory in Context, 2nd ed., Proposition 5.2.13 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter VI.3 (standard reference, not scraped)