TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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The endofunctor category of a small category is strict monoidal under composition

Statement

If $C$ is a small category, then the endofunctors of $C$ and their natural transformations form a strict monoidal category under functor composition, with tensor unit $1_{C}$ .

Facts & Assumptions

Given: A small category $C$ .

[L1]

When the source is small, functors $C \to C$ and natural transformations between them form the functor category $[C, C]$ (Functor category $[C, D]$ ).

[L2]

If both source and target are small, that functor category is itself small (If $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small).

[L3]

A strict monoidal category has literal associativity and unit equalities and identity constraints (Strict monoidal category).

Proof

technique · direct

1.1givenL1L2

By [L1] and [L2], $[C, C]$ is a legitimate category whose objects are endofunctors of $C$ and whose morphisms are natural transformations.

2.1step 1.1construct

Define the tensor product on objects by $F \otimes G : = F \circ G$ , and on morphisms by whiskered horizontal composition of natural transformations. The unit object is the identity functor $1_{C}$ .

3.1step 2.1L3

Composition of functors is literally associative and unital, so $((F \otimes G) \otimes H) = F \otimes (G \otimes H)$ and $1_{C} \otimes F = F = F \otimes 1_{C}$ on the nose. The associator and unitors are therefore identity transformations.

3.2step 2.1algebra

Whiskering respects identities and compositions, so the tensor on morphisms is a bifunctor.

4.1step 3.1step 3.2L3∎

Hence $[C, C]$ is strict monoidal under composition.

Depends on

Used by

Dependency tree · two levels

9 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Example 2.3.12 (standard reference, not scraped)
E. Riehl, Category Theory in Context, Chapter 5.1 (standard reference, not scraped)