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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Coalgebra homomorphisms are closed under identities and composition

Statement

For a comonad $G$ , every identity on a $G$ -coalgebra is a coalgebra homomorphism, and the composite of coalgebra homomorphisms is a coalgebra homomorphism.

Facts & Assumptions

Given: $G$ -coalgebras $(A, c)$ , $(B, d)$ , and $(C, e)$ .

[L1]

A coalgebra homomorphism $f : (A, c) \to (B, d)$ satisfies $G (f) c = d f$ (Coalgebra and coalgebra homomorphism for a comonad).

Proof

technique · direct

1.1L1

Functoriality gives $G (1_{A}) c = 1_{G A} c = c = c 1_{A}$ , so $1_{A}$ is a coalgebra homomorphism by [L1].

2.1L1step 1.1∎

If $f : (A, c) \to (B, d)$ and $g : (B, d) \to (C, e)$ are coalgebra homomorphisms, then $G (g f) c = G (g) G (f) c = G (g) d f = e g f$ . Thus $g f$ is a coalgebra homomorphism.

Depends on

Coalgebra and coalgebra homomorphism for a comonad

Used by

Co-Eilenberg–Moore category of a comonad Definition

Dependency tree · two levels

2 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., Exercise 5.2.iii (standard reference, not scraped)