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

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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 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 · two levels

8 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

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