PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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.

Every functor preserves isomorphisms

Statement

Every functor sends isomorphisms to isomorphisms.

Facts & Assumptions

Given: A functor $F : C \to D$ and an isomorphism $f : A \to B$ .

[L1]

Functors preserve identities and composition (Covariant functor, identity functor, composite functor, and contravariant functor), and an isomorphism has a two-sided inverse (Isomorphism, groupoid, and connected category).

Proof

technique · direct

1.1

Let $g : B \to A$ satisfy $g \circ f = 1_{A}$ and $f \circ g = 1_{B}$ .

givenL1

2.1

Functoriality gives $F (g) \circ F (f) = F (1_{A}) = 1_{F A}$ and $F (f) \circ F (g) = F (1_{B}) = 1_{F B}$ .

step 1.1L1

3.1

Thus $F (g)$ is a two-sided inverse of $F (f)$ , so $F (f)$ is an isomorphism.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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, Chapter 1 (standard reference, not scraped)