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 hom-assignment $C (-, -) : C^{o p} \times C \to S e t$ is a bifunctor

Statement

For every locally small category $C$ , the hom-assignment

$C (-, -) : C^{o p} \times C ⟶ S e t$

of The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category is a functor. Its restrictions in the two variables are the contravariant and covariant hom-functors of The assignments $C (a, -)$ and $C (-, a)$ are functors to $S e t$ .

Facts & Assumptions

Given: A locally small category $C$ ; morphisms $h : a^{'} \to a$ , $k : a^{''} \to a^{'}$ , $u : b \to b^{'}$ , and $v : b^{'} \to b^{''}$ ; and the category axioms of $C$ .

[L1]

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

[F1]

A morphism in a product category is a pair, and identities and composition are componentwise (Product category and its projection functors).

[F2]

The hom-assignment sends $(h, u)$ to $f \mapsto u \circ f \circ h$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

Proof

technique · direct

1.1

If $f : a \to b$ , then $u \circ f \circ h : a^{'} \to b^{'}$ , so [F2] defines a function $C (a, b) \to C (a^{'}, b^{'})$ .

givenF2

1.2

For every $f : a \to b$ , the identity pair acts by $1_{b} \circ f \circ 1_{a} = f$ .

givenF1F2

1.3

Applying $(h, u)$ and then $(k, v)$ sends $f$ to $v \circ (u \circ f \circ h) \circ k = (v \circ u) \circ f \circ (h \circ k)$ , which is the action of their componentwise composite in $C^{o p} \times C$ .

givenF1F2

2.1

Steps 1.1--1.3 prove the functor laws, and fixing either variable recovers the postcomposition or precomposition action of [L1]; hence the hom-assignment is the asserted bifunctor.

step 1.1step 1.2step 1.3L1∎

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: 12 results over 8 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

Tom Leinster, Basic Category Theory, Definition 4.1.22 and Remarks 4.1.23 (standard reference, not scraped)
Emily Riehl, Category Theory in Context, Chapter 2, Section 2.1 (standard reference, not scraped)

The hom-assignment C(−,−):Cop×C→SetC(−,−):Cop×C→Set is a bifunctor