ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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.
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The arrow category $\mathbf{Set}^{\to}$ : functions as objects and commuting squares as morphisms

Example

The functor category from the walking-arrow category to $\mathbf{Set}$ is the arrow category $\mathbf{Set}^{\to}$ .

Facts & Assumptions

Given: The walking-arrow category $\mathbf 2=(0\to1)$ .

[L1]

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

[L2]

Sets and functions form the category $\mathbf{Set}$ (Sets and functions form the large locally small category $\mathbf{Set}$ ).

Verification

technique · direct

1.1

Let $\mathbf 2$ have objects $0,1$ , their identities, and one further arrow $a:0\to1$ . A functor $F:\mathbf2\to\mathbf{Set}$ 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 $[\mathbf2,\mathbf{Set}]$ is precisely $\mathbf{Set}^{\to}$ as described.

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: 23 results over 11 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, Example 1.5.2 (standard reference, not scraped)

The arrow category Set→\mathbf{Set}^{\to}Set→: functions as objects and commuting squares as morphisms