ExampleConstruction: Literature-sourcedVerification: 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.

A Cartesian product represents $X \mapsto S e t (X, A) \times S e t (X, B)$

Example

For sets $A$ and $B$ , the presheaf

$F (X) = S e t (X, A) \times S e t (X, B)$

is represented by $A \times B$ . The representing natural isomorphism sends a function $f : X \to A \times B$ to its coordinate functions $(π_{A} \circ f, π_{B} \circ f)$ .

Facts & Assumptions

Given: Sets $A, B$ , the category $S e t$ , and an arbitrary set $X$ .

[F1]

A presheaf is represented by $R$ when it is naturally isomorphic to $S e t (-, R)$ (Presheaves, covariantly and contravariantly representable functors, and representations).

[F2]

Sets and functions form a locally small category whose composition and identities are ordinary function composition and identity functions (Sets and functions form the large locally small category $S e t$ ).

[F3]

The Cartesian product is $A \times B = {(a, b) : a \in A, b \in B}$ , and $(a, b) = (a^{'}, b^{'})$ holds exactly when $a = a^{'}$ and $b = b^{'}$ (The Cartesian product $A \times B : = {z \in P (P (A \cup B)) : \exists a \in A \exists b \in B z = (a, b)}$ , $(a, b) = (c, d)$ if and only if $a = c$ and $b = d$ ). Hence $π_{A} (a, b) = a$ and $π_{B} (a, b) = b$ are well-defined functions.

[F4]

A function assigns each element of its domain exactly one element of its codomain, and two functions with the same domain and codomain are equal exactly when their values agree everywhere (A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain, Functions $f$ and $g$ are equal if and only if $dom f = dom g$ and $f (x) = g (x)$ for every $x$ in that common domain).

Verification

technique · constructive

1.1

Define $Φ_{X} (f) = (π_{A} \circ f, π_{B} \circ f)$ for $f : X \to A \times B$ .

F2F3construct

1.2

For functions $g : X \to A$ and $h : X \to B$ , define $Ψ_{X} (g, h) : X \to A \times B$ by $Ψ_{X} (g, h) (x) = (g (x), h (x))$ .

F3F4construct

2.1

For every $(g, h)$ , the two projections of $Ψ_{X} (g, h)$ are $g$ and $h$ , so $Φ_{X} Ψ_{X} (g, h) = (g, h)$ .

step 1.1step 1.2F3F4

2.2

For every $f$ and $x \in X$ , $Ψ_{X} Φ_{X} (f) (x) = (π_{A} (f (x)), π_{B} (f (x))) = f (x)$ ; hence $Ψ_{X} Φ_{X} (f) = f$ .

step 1.1step 1.2F3F4

2.3

If $k : Y \to X$ , then $Φ_{Y} (f \circ k) = ((π_{A} \circ f) \circ k, (π_{B} \circ f) \circ k)$ , which is the restriction of $Φ_{X} (f)$ along $k$ in both factors. Thus $Φ$ is natural in $X$ .

step 1.1F2

3.1

Steps 2.1 and 2.2 make every $Φ_{X}$ bijective, and step 2.3 makes the family natural; by [F1], $A \times B$ represents $F$ .

step 2.1step 2.2step 2.3F1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 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

Justin Campbell, Harvard Math 55b tutorial notes, Example 1.1 and Definition 2.1 (standard reference, not scraped)

A Cartesian product represents X↦Set(X,A)×Set(X,B)X↦Set(X,A)×Set(X,B)