LemmaStatement: Literature-sourcedProof: 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.

Evaluation at the identity gives $Nat (C (a, -), F) ≅ F (a)$ and proves that the natural-transformation collection is a set

Statement

Let $C$ be a locally small category, let $a$ be an object, and let $F : C \to S e t$ be a functor. Evaluation at the identity defines a bijection

$E_{a, F} : Nat (C (a, -), F) ⟶ F (a), E_{a, F} (α) = α_{a} (1_{a}) .$

Its inverse sends $x \in F (a)$ to the natural transformation $α^{x}$ whose component at $c$ is

$α_{c}^{x} : C (a, c) ⟶ F (c), α_{c}^{x} (f) = F (f) (x) .$

The explicit parametrization by the set $F (a)$ proves that the natural-transformation collection in the display is a set. The construction makes no choice from a family of nonempty sets.

Facts & Assumptions

Given: A locally small category $C$ , an object $a$ , a functor $F : C \to S e t$ , and the identity, composition, and functor laws.

[F1]

The assignment $C (a, -)$ is the functor that sends $c$ to $C (a, c)$ and $u : c \to d$ to postcomposition $f \mapsto u \circ f$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category, The assignments $C (a, -)$ and $C (-, a)$ are functors to $S e t$ ).

[F2]

A natural transformation $α : H \Rightarrow F$ has components $α_{c} : H (c) \to F (c)$ satisfying $F (u) \circ α_{c} = α_{d} \circ H (u)$ for every $u : c \to d$ (Natural transformation and its components).

[F3]

A function is bijective when it is injective and surjective, equivalently when it has a two-sided inverse (Injection, surjection, bijection).

Proof

technique · constructive

1.1givenF1F2construct

For every natural transformation $α : C (a, -) \Rightarrow F$ , the value $α_{a} (1_{a})$ lies in $F (a)$ , so evaluation defines the displayed map $E_{a, F}$ .

1.2givenF1construct

For $x \in F (a)$ and each object $c$ , define $α_{c}^{x} (f) : = F (f) (x)$ for $f : a \to c$ .

2.1step 1.2F1F2

If $u : c \to d$ , then $F (u) (α_{c}^{x} (f)) = F (u) (F (f) (x)) = F (u \circ f) (x) = α_{d}^{x} (u \circ f)$ ; by [F1] and [F2], the family $α^{x}$ is natural.

2.2step 1.1step 1.2

Evaluation recovers $x$ : $E_{a, F} (α^{x}) = α_{a}^{x} (1_{a}) = F (1_{a}) (x) = x$ .

2.3step 1.1step 1.2F1F2

If $α : C (a, -) \Rightarrow F$ and $f : a \to c$ , naturality at $f$ gives $α_{c} (f) = α_{c} (f \circ 1_{a}) = F (f) (α_{a} (1_{a})) = α_{c}^{E_{a, F} (α)} (f)$ .

3.1step 2.1step 2.2step 2.3F3discharge-construct∎

Steps 2.2 and 2.3 make $x \mapsto α^{x}$ a two-sided inverse to $E_{a, F}$ , so [F3] gives the claimed bijection; its range is indexed by the set $F (a)$ , and every inverse value is given by a formula, so the asserted sethood and choice-freeness follow.

Evaluation at the identity gives $Nat (C (a, -), F) ≅ F (a)$ and proves that the natural-transformation collection is a set

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Evaluation at the identity gives Nat⁡(C(a,−),F)≅F(a)Nat(C(a,−),F)≅F(a) and proves that the natural-transformation collection is a set

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Evaluation at the identity gives $Nat (C (a, -), F) ≅ F (a)$ and proves that the natural-transformation collection is a set