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

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Horizontal and vertical composition of natural transformations satisfy the interchange law

Statement

Whenever the expressions are defined,

$(β^{'} \circ β) * (α^{'} \circ α) = (β^{'} * α^{'}) \circ (β * α) .$

Thus horizontal and vertical composition of natural transformations satisfy the interchange law.

Facts & Assumptions

Given: Natural transformations $α : F \Rightarrow G$ , $α^{'} : G \Rightarrow K : C \to D$ and $β : H \Rightarrow L$ , $β^{'} : L \Rightarrow M : D \to E$ .

[L1]

Vertical composition is componentwise and its composites are natural (Identity natural transformation and vertical composition, Vertical composites of natural transformations satisfy naturality); horizontal composition has the standard component formula and its composites are natural (Whiskering and horizontal composition of natural transformations, Horizontal composites of natural transformations satisfy naturality).

Proof

technique · direct

1.1

At an object $A$ , expand the left side by [L1] as $β_{K A}^{'} \circ β_{K A} \circ H (α_{A}^{'}) \circ H (α_{A})$ .

givenL1

2.1

Naturality of $β$ at $α_{A}^{'} : G A \to K A$ gives $β_{K A} \circ H (α_{A}^{'}) = L (α_{A}^{'}) \circ β_{G A}$ . Substitution turns step 1.1 into $(β_{K A}^{'} \circ L (α_{A}^{'})) \circ (β_{G A} \circ H (α_{A}))$ .

step 1.1L1

3.1

The two parenthesised factors are the $A$ -components of $β^{'} * α^{'}$ and $β * α$ , so every component agrees with the right side and the transformations are equal.

step 2.1L1∎

Depends on

Used by

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

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)