TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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

1.1

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 7 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, 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 · next 3 levels

Sources

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