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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Co-Kleisli composition is associative and unital

Statement

For a comonad $(G, ε, δ)$ , regard a co-Kleisli arrow $A ⇝ B$ as a morphism $f : G A \to B$ . Define

$g ⋆ f = g G (f) δ_{A} : G A \to C$

for $g : G B \to C$ , and define the identity at $A$ to be $ε_{A} : G A \to A$ . Then $⋆$ is associative and unital.

Facts & Assumptions

Given: A comonad $(G, ε, δ)$ and co-Kleisli arrows $f : G A \to B$ , $g : G B \to C$ , and $h : G C \to D$ .

[L1]

The comonad equations are coassociativity of $δ$ and the two counit laws for $ε$ (Comonad on a category).

[L2]

For a monad $(T, η, μ)$ and morphisms $f : A \to T B$ , $g : B \to T C$ , the composite $g ⋆ f : = μ_{C} \circ T (g) \circ f$ is associative, and $η_{A} : A \to T A$ is a two-sided identity at $A$ (Kleisli composition is associative and unital).

Proof

technique · direct

1.1

The formula $g ⋆ f = g G (f) δ_{A}$ has source $G A$ and target $C$ , so it defines composition on the proposed arrows.

2.1

Expanding $h ⋆ (g ⋆ f)$ and $(h ⋆ g) ⋆ f$ , naturality and coassociativity of $δ$ move the two duplications into the same order, after which functoriality of $G$ makes the composites equal.

L1L2step 1.1

3.1

Naturality of $ε$ at $f$ rewrites $ε_{B} ⋆ f = ε_{B} G (f) δ_{A}$ as $f ε_{G A} δ_{A}$ , so the equation $ε G \circ δ = 1_{G}$ gives $ε_{B} ⋆ f = f$ ; and $f ⋆ ε_{A} = f G (ε_{A}) δ_{A}$ , so the equation $G ε \circ δ = 1_{G}$ gives $f ⋆ ε_{A} = f$ . Hence $ε_{A}$ is a two-sided identity.

L1L2step 1.1∎

Depends on

Used by

Co-Kleisli category of a comonad Definition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 8 results over 7 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., Definition 5.2.10 by formal duality (standard reference, not scraped)