MATH.17:5.2 - Lift a rule, then change how repetition is distributed

Take integer operations f(n)=n+1 and g(n)=2n. Pointwise lifting to pairs is the case I={1,2}. Applied to (1,3), the lifted composite produces (4,8). Lifting f and g separately and then composing produces the same pair. The proof in :4.4 establishes that agreement for arbitrary inputs and functions.

Consider a different change: S(h)=h∘h, meaning repeat an operation twice. This always gives another integer endofunction, but the two sequencing proposals give:

S(g∘f)(n)=4n+6;

(S(g)∘S(f))(n)=4n+8.

At n=0 the answers are 6 and 8. To repeat the complete sequence, retain (g∘f)∘(g∘f). To run each stage twice, use g∘g∘f∘f. If f and g commute, rearrangement proves the two proposals equal; the present f and g do not.

The construction therefore returns both a valid operation on operations and a failed composition-preservation claim. That failure determines which changed rule implements the intended repetition.