PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 $F:\mathcal C\to\mathcal D$ 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 $F(g)\circ F(f)=F(1_A)=1_{FA}$ and $F(f)\circ F(g)=F(1_B)=1_{FB}$ .

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 4 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

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)