CorollaryStatement: AI-adaptedProof: Literature-sourcedprecheck 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

1.1L1

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

1.2L2

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

2.1step 1.1step 1.2∎

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.

Depends on

Used by

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

Dependency tree · two levels

7 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)