LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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Horizontal composites of natural transformations satisfy naturality

Statement

The horizontal composite of natural transformations satisfies the naturality equation.

Facts & Assumptions

Given: Natural transformations $α : F \Rightarrow G : C \to D$ and $β : H \Rightarrow L : D \to E$ , and $f : A \to B$ in $C$ .

[L1]

Horizontal composition has component $β_{G A} \circ H (α_{A}) = L (α_{A}) \circ β_{F A}$ , and functors preserve composition (Whiskering and horizontal composition of natural transformations).

Proof

technique · direct

1.1givenL1

Naturality of $β$ at $G f$ gives $L (G f) \circ β_{G A} = β_{G B} \circ H (G f)$ , while functoriality sends the naturality equation $G f \circ α_{A} = α_{B} \circ F f$ to $H (G f) \circ H (α_{A}) = H (α_{B}) \circ H (F f)$ .

2.1step 1.1L1

Combining these equations gives $L (G f) \circ β_{G A} \circ H (α_{A}) = β_{G B} \circ H (α_{B}) \circ H (F f)$ .

3.1step 2.1L1∎

The two outer composites are exactly $L (G f) \circ (β * α)_{A}$ and $(β * α)_{B} \circ H (F f)$ , which proves naturality.

Horizontal composites of natural transformations satisfy naturality

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources