CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-29

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.

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Two morphisms agreeing on every member need not be equal

Statement refuted

If two morphisms $f, g : A \to B$ satisfy $f x \equiv g x$ for every member $x$ of $A$ , then $f = g$ .

Facts & Assumptions

Given: The abelian category $A b$ and the two endomorphisms $1_{Z}, - 1_{Z} : Z \to Z$ .

[L1]

The category $A b$ is abelian (Abelian groups form an abelian category).

[L2]

Every member has a negative, and equivalence to zero is literal equality (Each object has a zero member and each member has a negative).

Counterexample

technique · direct

1.1L1L2algebra

Let $x : X \to Z$ be any member. Then $(- 1_{Z}) x = x (- 1_{X})$ , and the automorphism $- 1_{X}$ is epic. Therefore $1_{Z} x \equiv (- 1_{Z}) x$ .

1.2L1algebra

Nevertheless $1_{Z} \neq - 1_{Z}$ , since they send $1 \in Z$ to different integers. So memberwise equivalence of composites does not force equality of the morphisms themselves.

2.1step 1.1step 1.2∎

This refutes the statement.

Depends on

Used by

FALSE: two morphisms that agree on every member are equal False statement

Dependency tree · two levels

12 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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)