TheoremStatement: AI-adaptedProof: Literature-sourcedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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The Kleisli factorisation functor for an adjunction inducing a monad exists and is unique

Statement

Let $F : C ⇄ D : U$ be an adjunction with counit $ε$ whose induced monad is a fixed monad $T$ on $C$ on the nose, and write $F_{T} ⊣ U_{T}$ for the Kleisli adjunction with counit $ε^{T}$ . There is exactly one functor $J : C_{T} \to D$ satisfying

$J F_{T} = F, U J = U_{T}, J (ε_{B}^{T}) = ε_{F B} for every object B,$

namely

$J (A) = F (A), J (f : A \to T B) = ε_{F B} \circ F (f) : F A \to F B .$

These three equalities are what it means for $J$ to be a morphism of adjunctions from the Kleisli adjunction to $F ⊣ U$ .

Facts & Assumptions

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

Proof

technique · direct

1.1given

A Kleisli arrow $f : A \to T B = U F B$ has adjoint transpose $ε_{F B} \circ F (f) : F A \to F B$ ; define $J$ by this formula and by $J (A) = F A$ .

2.1step 1.1given

For an identity $η_{A}$ , the first triangle identity gives $J (η_{A}) = ε_{F A} \circ F (η_{A}) = 1_{F A}$ . For $f : A \to T B$ and $g : B \to T C$ , substitution of $μ = U ε F$ , naturality of $ε$ , and the triangle identities gives $J (g ⋆ f) = J (g) \circ J (f)$ , so $J$ is a functor.

3.1step 1.1step 2.1∎

The formulas give $J F_{T} = F$ and $U J = U_{T}$ on objects and arrows, and $J$ sends the Kleisli counit to $ε$ . Conversely, those equalities force the image of $f : A \to U F B$ to be its adjoint transpose, so they force the formula of step 1.1 on every arrow and prove uniqueness.

Depends on

Used by

Dependency tree · two levels

11 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

E. Riehl, Category Theory in Context, 2nd ed., Proposition 5.2.13 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Theorem 6.3.10 (standard reference, not scraped)