ExampleConstruction: Literature-sourcedVerification: 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.

The arrow category ${S e t}^{\to}$ : functions as objects and commuting squares as morphisms

Example

The functor category from the walking-arrow category to $S e t$ is the arrow category ${S e t}^{\to}$ .

Facts & Assumptions

Given: The walking-arrow category $2 = (0 \to 1)$ .

[L1]

Objects and morphisms in a functor category are functors and natural transformations (Functor category $[C, D]$ ).

[L2]

Sets and functions form the category $S e t$ (Sets and functions form the large locally small category $S e t$ ).

Verification

technique · direct

1.1

Let $2$ have objects $0, 1$ , their identities, and one further arrow $a : 0 \to 1$ . A functor $F : 2 \to S e t$ is exactly a function $F (a) : F (0) \to F (1)$ .

L1L2

2.1

Given functions $f : X \to Y$ and $g : X^{'} \to Y^{'}$ , a natural transformation between their corresponding functors consists of maps $u : X \to X^{'}$ and $v : Y \to Y^{'}$ satisfying $v f = g u$ . Thus it is exactly a commuting square.

step 1.1L1

3.1

Vertical composition composes the two side maps of commuting squares. It preserves the square equation, and identity transformations give identity squares. Therefore $[2, S e t]$ is precisely ${S e t}^{\to}$ as described.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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, Example 1.5.2 (standard reference, not scraped)

The arrow category Set→Set→: functions as objects and commuting squares as morphisms