TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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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 assignments $C (a, -)$ and $C (-, a)$ are functors to $S e t$

Statement

Let $C$ be a locally small category and let $a$ be an object. The covariant hom-assignment $C (a, -) : C \to S e t$ and the contravariant hom-assignment $C (-, a) : C^{o p} \to S e t$ of The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category are functors.

Facts & Assumptions

Given: A locally small category $C$ , an object $a$ , morphisms $u : b \to c$ and $v : c \to d$ , and the identity and associativity axioms of $C$ .

[F1]

The covariant assignment sends $u : b \to c$ to $u_{*} : f \mapsto u \circ f$ , while the contravariant assignment sends it to $u^{*} : g \mapsto g \circ u$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

[F2]

A covariant functor preserves identities and composites; a contravariant functor from $C$ means a covariant functor from $C^{o p}$ (Covariant functor, identity functor, composite functor, and contravariant functor).

Proof

technique · direct

1.1givenF1

The maps in [F1] have the asserted source and target: if $f : a \to b$ then $u \circ f : a \to c$ , and if $g : c \to a$ then $g \circ u : b \to a$ .

1.2givenF1

Postcomposition by $1_{b}$ fixes every $f : a \to b$ , and precomposition by $1_{b}$ fixes every $g : b \to a$ .

1.3givenF1

For $f : a \to b$ , one has $(v \circ u)_{*} (f) = (v \circ u) \circ f = v \circ (u \circ f) = v_{*} (u_{*} (f))$ .

1.4givenF1

For $g : d \to a$ , one has $(v \circ u)^{*} (g) = g \circ (v \circ u) = (g \circ v) \circ u = u^{*} (v^{*} (g))$ , which is the composition law in $C^{o p}$ .

2.1step 1.1step 1.2step 1.3step 1.4F2∎

Steps 1.1--1.4 give the identity and composition laws required by [F2], so both hom-assignments are functors to $S e t$ .

The assignments $C (a, -)$ and $C (-, a)$ are functors to $S e t$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The assignments C(a,−)C(a,−) and C(−,a)C(−,a) are functors to SetSet

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The assignments $C (a, -)$ and $C (-, a)$ are functors to $S e t$