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

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Lax monoidal functors compose, and composition preserves strength and strictness

Statement

The composite of two lax monoidal functors is again lax monoidal. If both functors are strong, the composite is strong; if both are strict, the composite is strict.

Facts & Assumptions

Given: Lax monoidal functors $F : C \to D$ and $G : D \to E$ .

[L1]

A lax monoidal functor is a functor together with structure maps $F_{2}, F_{0}$ satisfying associativity and unit equations; strong means those maps are isomorphisms and strict means they are identities (Lax, strong, and strict monoidal functors).

[L2]

A monoidal natural transformation is compatible with the binary and unit structure maps (Monoidal natural transformation).

Proof

technique · direct

1.1

Define the composite structure maps by $(G F)_{0} : = G (F_{0}) \circ G_{0}$ and $(G F)_{2; X, Y} : = G (F_{2; X, Y}) \circ G_{2; F (X), F (Y)}$ . These are the only typed composites from $G (F (X)) \otimes G (F (Y))$ to $G F (X \otimes Y)$ and from the unit of $E$ to $G F (1)$ .

givenL1construct

2.1

Paste the associativity square for $G$ with the image under $G$ of the associativity square for $F$ . The outside rectangle is exactly the associativity axiom for $(G F)_{2}$ . Pasting the two unit squares gives the left and right unit axioms for $(G F)_{0}$ . Hence $G F$ is lax monoidal.

step 1.1L1L2

3.1

If both $F$ and $G$ are strong, then every map used in step 1.1 is an isomorphism, so $(G F)_{0}$ and $(G F)_{2}$ are isomorphisms. If both are strict, those maps are identities, so the composite structure maps are identities too.

step 1.1step 2.1L1

4.1

Therefore composition preserves laxness, strength, and strictness.

step 2.1step 3.1∎

Depends on

Used by

Monoidal equivalence and monoidal quasi-inverse data Definition

Dependency tree · two levels

6 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, Chapter 2.4 (standard reference, not scraped)