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.
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.

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

1.1

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 $g \circ u : b \to a$ .

givenF1

1.2

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

givenF1

1.3

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

givenF1

1.4

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 $C^{o p}$ .

givenF1

2.1

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

step 1.1step 1.2step 1.3step 1.4F2∎

Depends on

Used by

Cited to discharge well-definedness by The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category.

Dependency tree · two levels

7 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

Emily Riehl, Category Theory in Context, Chapter 2, Section 2.1 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Definitions 4.1.1 and 4.1.16 (standard reference, not scraped)

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

Statement

Facts & Assumptions

Proof

Depends on

Used by

Dependency tree · two levels

Sources

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