TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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.

Initial and terminal objects are unique up to a unique isomorphism

Statement

Any two initial objects in a category are joined by a unique isomorphism. Any two terminal objects are likewise joined by a unique isomorphism.

Facts & Assumptions

Given: A category $C$ and either two initial objects $I, I^{'}$ or two terminal objects $T, T^{'}$ .

[F1]

From an initial object there is exactly one morphism to every object, and into a terminal object there is exactly one morphism from every object (Initial object, terminal object, and zero object).

[F2]

A morphism is an isomorphism when it has a two-sided inverse (Isomorphism, groupoid, and connected category).

[L1]

A formal theorem derived from the category axioms has a dual obtained by reversing morphisms and composition (Every theorem about categories has a formal dual obtained by reversing morphisms and composition).

Proof

technique · direct

1.1

If $I$ and $I^{'}$ are initial, let $f : I \to I^{'}$ and $g : I^{'} \to I$ be their unique morphisms in the indicated directions.

givenF1

1.2

The identity is the unique endomorphism of an initial object, so $g \circ f = 1_{I}$ and $f \circ g = 1_{I^{'}}$ .

2.1

By steps 1.1 and 1.2, $f$ is an isomorphism with inverse $g$ .

step 1.1step 1.2F2

2.2

Any isomorphism $I \to I^{'}$ is a morphism with that source and target and must equal the unique morphism $f$ , so the isomorphism is unique.

step 1.1F1

3.1

Applying the dual argument of [L1] to terminal objects gives a unique isomorphism $T \to T^{'}$ .

step 2.1step 2.2L1∎

Depends on

Used by

Dependency tree · two levels

5 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

Emily Riehl, Category Theory in Context, Chapters 1 and 2 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Chapter 4 (standard reference, not scraped)