LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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

1.1

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 $E_{a, F}$ .

givenF1F2construct

1.2

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

givenF1construct

2.1

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.

step 1.2F1F2

2.2

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

step 1.1step 1.2

2.3

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

step 1.1step 1.2F1F2

3.1

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.

step 2.1step 2.2step 2.3F3discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 10 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, Theorem 2.2.4 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Theorem 4.2.1 (standard reference, not scraped)

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