CorollaryStatement: 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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Kleisli and Eilenberg–Moore adjunctions have the extremal universal properties

Statement

Fix a monad $T$ on $C$ . Among supplied adjunctions inducing $T$ on the nose, the Kleisli adjunction has the schematic initial universal property and the Eilenberg–Moore adjunction has the schematic terminal universal property: every such adjunction admits a unique morphism of adjunctions from the Kleisli resolution and a unique morphism of adjunctions to the Eilenberg–Moore resolution.

Facts & Assumptions

Given: A fixed monad $T$ and an arbitrary supplied adjunction inducing it.

[L1]

For an adjunction $F ⊣ U$ with counit $ε$ inducing $T$ on the nose there is exactly one functor $J : C_{T} \to D$ with $J F_{T} = F$ , $U J = U_{T}$ and $J (ε_{B}^{T}) = ε_{F B}$ for every $B$ (The Kleisli factorisation functor for an adjunction inducing a monad exists and is unique).

[L2]

For the same adjunction there is exactly one functor $K : D \to C^{T}$ with $U^{T} K = U$ , $K F = F^{T}$ and $K (ε_{d}) = ε_{K d}^{T}$ for every $d$ (The comparison functor to the Eilenberg–Moore category exists and is unique).

Proof

technique · direct

1.1

Apply [L1] to the supplied adjunction; its unique factorisation is the required morphism from the Kleisli resolution.

1.2

Apply [L2] to the same adjunction; its unique comparison is the required morphism to the Eilenberg–Moore resolution.

2.1

Steps 1.1 and 1.2 give the two asserted existence-and-uniqueness properties for each supplied adjunction. This objectwise assertion does not form a category of all resolutions.

step 1.1step 1.2∎

Depends on

Used by

Co-Kleisli and co-Eilenberg–Moore adjunctions have the dual extremal universal properties Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 6 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)
B. Richter, From Categories to Homotopy Theory, Theorem 6.3.10 (standard reference, not scraped)